Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^d$ and risk-sensitive control

TL;DR

This paper investigates the principal eigenvalues of elliptic operators in ^d, linking their strict monotonicity to ergodic properties and applying these insights to risk-sensitive control problems for diffusions.

Contribution

It characterizes the strict monotonicity of principal eigenvalues in relation to ergodic properties and extends results to equations with measurable coefficients, also establishing duality in ergodic control.

Findings

Strict monotonicity characterizes ergodic properties and uniqueness of ground states.

Established strong duality for ergodic control linear programming formulations.

Proved existence and optimality of Markov controls in risk-sensitive control problems.

Abstract

This paper studies the eigenvalue problem on $\mathbb{R}^d$ for a class of second order, elliptic operators of the form $\mathscr{L} = a^{ij}\partial_{x_i}\partial_{x_j} + b^{i}\partial_{x_i} + f$, associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function $f$ fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.

Equations

$${\mathcal{L}}^f\phi =\sum _{i,j=1}^d{a}^{ij}\frac{{\partial }^2\phi }{\partial x_i\partial x_j}+\sum _{i=1}^d{b}^i\frac{\partial \phi }{\partial x_i}+f\phi .$$

$$X_t=x+{\int }_0^{t}b(X_s)\mathrm{d}s+{\int }_0^t\text{upsigma}(X_s)\mathrm{d}{W}_s,\text{with}a:=\frac{1}{2}\text{upsigma}{\text{upsigma}}^{\mathsf{T}},$$

$$\mathscr{E}_{x}(f)\;\coloneqq\;\limsup_{T\to\infty}\,\frac{1}{T}\;\log\,\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{T}f(X_{s})\,\mathrm{d}{s}}\Bigr{]}\,,\quad x\in{\mathbb{R}^{d}}\,,$$

$${\mathcal{L}}^f\Psi =a^{ij}{\partial }_{ij}\Psi +b^i{\partial }_i\Psi +f\Psi =\lambda \Psi .$$

$$a^{ij}\partial_{ij}\breve{\psi}+b^{i}\partial_{i}\breve{\psi}-\langle\nabla\breve{\psi},a\nabla\breve{\psi}\rangle\;=\;a^{ij}\partial_{ij}\breve{\psi}+b^{i}\partial_{i}\breve{\psi}+\min_{u\in{\mathbb{R}^{d}}}\bigl{[}2\langle a\,u,\nabla\breve{\psi}\rangle+\langle u,au\rangle\bigr{]}\;=\;f-\lambda^{\!*}(f)\,.$$

$$\alpha_{*}\;=\;\biggl{\{}\inf\;\int_{{\mathbb{R}^{d}}\times\mathbb{U}}\mathscr{R}(x,u)\,\uppi(\mathrm{d}{x},\mathrm{d}{u})\;\colon\;\mathcal{A}^{*}\uppi=0\,,\ \ \uppi\in\mathcal{P}({\mathbb{R}^{d}}\times\mathbb{U})\biggr{\}}\,.$$

$$\alpha\;=\;\sup\;\Bigl{\{}c\in\mathbb{R}\;\colon\;\inf_{u\in\mathbb{U}}\,\bigl{[}\mathcal{A}g(x,u)+\mathscr{R}(x,u)\bigr{]}\geq c\,,\ \ g\in\mathcal{D}(\mathcal{A})\Bigr{\}}\,,$$

$$\Vert \text{upsigma}(x)-\text{upsigma}(y)\Vert \le C_R|x-y|\forall x,y\in B_R.$$

$$\langle b(x),x\rangle^{+}+\lVert\upsigma(x)\rVert^{2}\;\leq\;C_{0}\bigl{(}1+\lvert x\rvert^{2}\bigr{)}\qquad\forall\,x\in\mathbb{R}^{d},$$

$$\sum _{i,j=1}^d{a}^{ij}(x){\xi }_i{\xi }_j\ge C_R^{-1}|\xi {|}^2\forall x\in B_R,$$

$$\text{uptau}(A):=\inf \{t>0:X_t\notin A\}.$$

$$⟨f,\mu ⟩=\mu (f):={\int }_{{\mathbb{R}}^d}f(x)\mu (\mathrm{d}x).$$

$$X_t=X_0+{\int }_0^{t}b(X_s)\mathrm{d}s+{\int }_0^t\text{upsigma}(X_s)\mathrm{d}{W}_s.$$

$$\mathcal{L}g(x):=a^{ij}(x){\partial }_{ij}g(x)+b^i(x){\partial }_{i}g(x).$$

$$\mathcal{L}{\hat{\Psi }}_r(x)+f(x){\hat{\Psi }}_r(x)={\hat{\lambda }}_r{\hat{\Psi }}_r(x)\text{a.e. }x\in B_r,$$

$$\mathscr{E}_{x}(f)\;\coloneqq\;\limsup_{T\to\infty}\,\frac{1}{T}\,\log\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{T}f(X_{s})\,\mathrm{d}{s}}\Bigr{]}\,,\quad\text{and}\quad\mathscr{E}(f)\;\coloneqq\;\inf_{x\in{\mathbb{R}^{d}}}\;\mathscr{E}_{x}(f).$$

$$\hat{\Lambda}(f)\;=\;\inf\,\bigl{\{}\lambda\in\mathbb{R}\;\colon\;\exists\,\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}}),\,\varphi>0,\,\mathscr{L}\varphi+(f-\lambda)\varphi\leq 0,\;\text{a.e. in}\;{\mathbb{R}^{d}}\bigr{\}}\,.$$

$$\widehat{\Psi}_{n}(x)\;=\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}_{r}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\widehat{\Psi}_{n}(X_{\breve{\uptau}_{r}})\,\mathds{1}_{\{\breve{\uptau}_{r}<\uptau_{n}\}}\Bigr{]}\qquad\forall\,x\in B_{n}\setminus\overline{B}_{r}\,,$$

$$\Psi^{*}(x)\;=\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\lambda^{\!*}(f)]\,\mathrm{d}{t}}\,\Psi^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\infty\}}\Bigr{]}\qquad\forall\,x\in\mathscr{B}^{c}\,.$$

$$\mathcal{L}\Psi +f\Psi ={\lambda }^{\ast }(f)\Psi \text{a.e.leavevmodenobreak on }{\mathbb{R}}^d.$$

$$\mathcal{L}\phi +(f-\lambda )\phi \le 0,\text{and}\lambda \ge \hat{\Lambda }(f).$$

$$\varphi(x)\;\geq\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}_{r}}[f(X_{t})-\lambda]\,\mathrm{d}{t}}\,\varphi(X_{\breve{\uptau}_{r}})\,\mathds{1}_{\{\breve{\uptau}_{r}<\infty\}}\Bigr{]}\,.$$

$$\mathcal{L}(\kappa \phi -{\hat{\Psi }}_r)-(f-{\hat{\lambda }}_r{)}^{-}(\kappa \phi -{\hat{\Psi }}_r)\le -(f-{\hat{\lambda }}_r{)}^{+}(\kappa \phi -{\hat{\Psi }}_r)+(-{\hat{\lambda }}_r+\lambda )\kappa \phi \le 0\text{in }{B}_r.$$

$$\Psi^{*}(x)\;\geq\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\lambda^{\!*}(f)]\,\mathrm{d}{t}}\,\Psi^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\infty\}}\Bigr{]}\,.$$

$$\tilde{\Psi}^{*}(x)\;\geq\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-h(X_{t})-\lambda^{\!*}(f-h)]\,\mathrm{d}{t}}\,\tilde{\Psi}^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\infty\}}\Bigr{]}\,,$$

$$\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-h(X_{t})-\lambda^{\!*}(f-h)]\,\mathrm{d}{t}}\,\mathds{1}_{\{\breve{\uptau}<\infty\}}\Bigr{]}\;<\;\infty\qquad\forall\;x\in\mathscr{B}^{c}\,,$$

$$\widehat{\Psi}_{n}(x)\;\leq\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\Psi^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\uptau_{n}\}}\Bigr{]}\;+\;\biggl{(}\sup_{\mathscr{B}}\,\bigl{\lvert}\Psi^{*}-\widehat{\Psi}_{n}\bigr{\rvert}\biggr{)}\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\mathds{1}_{\{\breve{\uptau}<\uptau_{n}\}}\Bigr{]}\,.$$

$$\kappa_{n}\;\coloneqq\;\Bigl{(}\inf_{\mathscr{B}}\,\widehat{\Psi}_{n}\Bigr{)}^{-1}\sup_{\mathscr{B}}\,\bigl{\lvert}\Psi^{*}-\widehat{\Psi}_{n}\bigr{\rvert}\,.$$

$$\biggl{(}\sup_{\mathscr{B}}\,\bigl{\lvert}\Psi^{*}-\widehat{\Psi}_{n}\bigr{\rvert}\biggr{)}\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\mathds{1}_{\{\breve{\uptau}<\uptau_{n}\}}\Bigr{]}\;\leq\;\kappa_{n}\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\widehat{\Psi}_{n}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\uptau_{n}\}}\Bigr{]}\;=\;\kappa_{n}\;\widehat{\Psi}_{n}(x)\,.$$

$$\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\hat{\lambda}_{n}]\,\mathrm{d}{t}}\,\Psi^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\uptau_{n}\}}\Bigr{]}\;\xrightarrow[n\to\infty]{}\;\operatorname{\mathbb{E}}_{x}\Bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\lambda^{\!*}(f)]\,\mathrm{d}{t}}\,\Psi^{*}(X_{\breve{\uptau}})\,\mathds{1}_{\{\breve{\uptau}<\infty\}}\Bigr{]}\,,$$

Full text

††journal: Journal de Mathématiques Pures et Appliquées

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1. 1 Introduction

2. 1.1 Assumptions on the model

3. 1.2 Notation

4. 2 General results

5. 2.1 Risk-sensitive value and Dirichlet eigenvalues

6. 2.2 Summary of results

7. 2.3 Proof of Theorem 2.1 and other results

8. 2.3.1 Minimal growth at infinity

9. 2.4 Potentials $f$ vanishing at infinity

10. 2.4.1 Strong duality

11. 2.4.2 Differentiability of $\Lambda_{\beta}$

12. 3 Exponential ergodicity and strict monotonicity of principal eigenvalues

13. 4 Risk-sensitive control

14. 4.1 The controlled diffusion model

15. 4.2 Relaxed controls

16. 4.3 Optimal Markov controls and the risk-sensitive HJB

17. 4.4 Continuity results

1 Introduction

In this paper we study the eigenvalue problem on ${\mathbb{R}^{d}}$ for non-degenerate, second order elliptic operators $\mathscr{L}^{f}$ of the form

[TABLE]

Here $b,f\in L_{\mathrm{loc}}^{\infty}({\mathbb{R}^{d}})$, $a\in C_{\mathrm{loc}}^{0,1}({\mathbb{R}^{d}})$ and $a$, $b$ satisfy a linear growth assumption in the outward radial direction (see (A2) in Subsection 1.1). In other words, $a$ and $b$ satisfy the usual assumptions for existence and uniqueness of a strong solution of the Itô equation

[TABLE]

where $W$ is a standard Brownian motion.

We focus on certain properties of the principal eigenvalue of the operator $\mathscr{L}^{f}$ which play a key role in infinite horizon risk-sensitive control problems. When $D$ is a smooth bounded domain, and $a$, $b$, $f$ are regular enough, existence of a principal eigenvalue and corresponding eigenfunction under a Dirichlet boundary condition can be obtained by an application of Krein-Rutman theory (see for instance [1, 2]). This eigenvalue is the bottom of the spectrum of $-\mathscr{L}^{f}$ with Dirichlet boundary condition. For non-smooth domains, a generalized notion of a principal eigenvalue was introduced in the seminal work of Berestycki, Nirenberg and Varadhan [3]. An analogous theory for non-linear elliptic operators has been developed by Quaas and Sirakov in [4]. The principal eigenvalue plays a key role in the study of non-homogeneous elliptic operators and the maximum principle (see [3, 5, 6, 4]). For some other definitions of the principal (or critical) eigenvalue we refer the reader to the works of Pinchover [7] and Pinsky [2, Chapter 4].

For unbounded domains, principal eigenvalue problems have been recently considered by Berestycki and Rossi in [8, 5]. Not surprisingly, certain properties of the principal eigenvalue which hold in bounded domains may not be true for unbounded ones. For instance, when $D$ is smooth and bounded it is well known that for the Dirichlet boundary value problem, the principal eigenvalue is simple, and the associated principal eigenfunction is positive. Moreover, it is the unique eigenvalue with a positive eigenfunction. But if $D$ is unbounded and smooth, then there exists a constant $\lambda^{\!*}=\lambda^{\!*}(f)$ such that any $\lambda\in[\lambda^{\!*},\infty)$ is an eigenvalue of $\mathscr{L}^{f}$ with a positive eigenfunction [5, Theorem 1.4] (see also [6] and [9, Theorem 2.6]). The lowest such value $\lambda^{\!*}$ serves as a definition of the principal eigenvalue when $D$ is not bounded. The principal eigenvalue is known to be strictly monotone as a function of the bounded domain $D$ (the latter ordered with respect to set inclusion), and also strictly monotone in the coefficient $f$ when the domain is bounded (see [5] and Lemma 2.1 below). These properties fail to hold in unbounded domains as remarked by Berestycki and Rossi [5, Remark 2.4]. Strict monotonicity of $f\mapsto\lambda^{\!*}(f)$ and its implications are a central theme in our study. We adopt a probabilistic approach in our investigation. One can view $\lambda^{\!*}(f)$ as a risk-sensitive average of $f$ over the diffusion in Eq. 1.2. More precisely, since Eq. 1.2 has a unique solution which exists for all $t\in[0,\infty)$, then we can define

[TABLE]

with ‘$\log$’ denoting the natural logarithm. As shown in the proof of Lemma 2.3 in [10] we have $\lambda^{\!*}(f)\leq\mathscr{E}_{x}(f)$, and equality is indeed the case in many important situations, although strictly speaking it is only a heuristic. This heuristic is based on the fact that for a bounded $f$, the operator $\mathscr{L}^{f}$ is the infinitesimal generator of a strongly continuous, positive semigroup with potential $f$, see for instance [11, Chapter IV]. If $f$ is a bounded continuous function, and if the occupation measures of $\{X_{t}\}$ obey a large deviation principle, then one can express $\mathscr{E}_{x}(f)$ in terms of the large deviation rate function. This is known as the variational representation for the eigenvalue. See for instance the article by Donsker and Varadhan [12] where this representation is obtained for compact domains. But large deviation principles for $\{X_{t}\}$ are generally available only under strong hypotheses on the process (see [13]). In this paper we rely on the stochastic representation of the principal eigenfunction which can be established under very mild assumptions. This approach has been recently used by Arapostathis and Biswas in [10] to study the multiplicative Poisson equation when $f$ is near-monotone (which includes the case of inf-compact $f$). By an eigenpair of $\mathscr{L}^{f}$ we mean a pair $(\Psi,\lambda)$, with $\Psi$ a positive function in $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, for all $p\in[1,\infty)$, and $\lambda\in\mathbb{R}$, that satisfies

[TABLE]

We refer to $\lambda$ as the eigenvalue, and to $\Psi$ as the eigenfunction. In Eq. 1.4, and elsewhere in this paper, we adopt the notation $\partial_{i}\coloneqq\tfrac{\partial\leavevmode\nobreak\ }{\partial{x}_{i}}$ and $\partial_{ij}\coloneqq\tfrac{\partial^{2}\leavevmode\nobreak\ }{\partial{x}_{i}\partial{x}_{j}}$ for $i,j\in\mathbb{N}$, and use the standard summation rule that repeated subscripts and superscripts are summed from $1$ through $d$.

As mentioned earlier, such a pair $(\Psi,\lambda)$ exists only if $\lambda\geq\lambda^{\!*}(f)$ (see Corollary 2.1). Given an eigenpair $(\Psi,\lambda)$, the associated twisted diffusion $Y$ (a terminology used in [14]) is an Itô process as in Eq. 1.2, but with the drift $b$ replaced by $b+2a\nabla(\log\Psi)$. It is not generally the case that the twisted process has a strong solution which exists for all time. If $\lambda>\lambda^{\!*}(f)$ the twisted diffusion is always transient (see Lemma 2.6). When $\lambda=\lambda^{\!*}(f)$, the eigenfunction is denoted as $\Psi^{*}$ and is called the ground state [2, 15]. The corresponding twisted diffusion, denoted by $Y^{*}$, is referred to as the ground-state diffusion.

Let $C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ ($C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$) denote the class of non-zero, nonnegative real valued continuous functions on ${\mathbb{R}^{d}}$ which vanish at infinity (have compact support). We say that $\lambda^{\!*}(f)$ is strictly monotone at $f$ if there exists $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ satisfying $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$. We also say that $\lambda^{\!*}(f)$ is strictly monotone at $f$ on the right if $\lambda^{\!*}(f+h)>\lambda^{\!*}(f)$ for all $h\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$. In Theorem 2.1 we show that strict monotonicity at $f$ implies strict monotonicity at $f$ on the right. Our main results provide sharp characterizations of the ground state $\Psi^{*}$ and the ground state process $Y^{*}$ in terms of these monotonicity properties. Assume that $f\colon{\mathbb{R}^{d}}\to\mathbb{R}$ is a locally bounded, Borel measurable function, satisfying $\operatorname*{ess\,inf}_{\mathbb{R}^{d}}f>-\infty$, and that $\lambda^{\!*}(f)$ is finite. We show that strict monotonicity of $\lambda^{\!*}(f)$ at $f$ on the right implies the simplicity of $\lambda^{\!*}(f)$, i.e., the uniqueness of the ground state $\Psi^{*}$, and that this is also a necessary and sufficient condition for the ground state process to be recurrent (see Lemmas 2.7 and 2.3). Another important result is that the ground state diffusion is exponentially ergodic (see Definition 2.2) if and only if $\lambda^{\!*}(f)$ is strictly monotone at $f$. These results are summarized in Theorem 2.1 in Section 2. Other results in Section 2 provide a characterization of the eigenvalue in terms of the long time behavior of the twisted process and stochastic representations of the ground state (see Lemmas 2.2, 2.3 and 2.7, and Theorem 2.6).

In [2], Pinsky uses the existence of a Green’s measure to define the critical eigenvalue of a non-degenerate elliptic operator. This critical eigenvalue coincides with the principal eigenvalue when the boundary of the domain and the coefficients of $\mathscr{L}^{f}$ are smooth enough. He shows that for any bounded domain, and provided that the coefficients are in $C^{1,\alpha}({\mathbb{R}^{d}})$, $\alpha>0$, and bounded, there exists a critical value $\lambda_{c}$ such that for any $\lambda>\lambda_{c}$ we can find a Green’s measure corresponding to the operator $\mathscr{L}^{(f-\lambda)}$ [2, Theorem 4.7.1]. The result in Theorem 2.3 in Section 2 extends this to ${\mathbb{R}^{d}}$ without assuming much regularity on the coefficients.

Continuous dependence of $\lambda^{\!*}$ on the coefficients of $\mathscr{L}$ has also been a topic of interest. It is not hard to see that $f\mapsto\lambda^{\!*}(f)$ is lower-semicontinuous in the $L_{\mathrm{loc}}^{1}({\mathbb{R}^{d}})$ topology for $f$. Continuity of this map is also established in [5, Proposition 9.2] with respect to the $L^{\infty}({\mathbb{R}^{d}})$ norm of $f$. In Theorems 2.6 and 4.1 we study the continuity of $\lambda^{\!*}(f)$ for a class of functions $f$ under the $L_{\mathrm{loc}}^{1}({\mathbb{R}^{d}})$ topology. We also obtain a pinned multiplicative ergodic theorem which is of independent interest, and show that $\mathscr{E}_{x}(f)=\lambda^{\!*}(f)$ for a large class of problems.

We next discuss the connection of this problem with a stochastic ergodic control problem. Defining $\breve{\psi}\coloneqq\log\Psi^{*}$ we obtain from Eq. 1.4 that

[TABLE]

It is easy to see that Eq. 1.5 is related to an ergodic control problem with controlled drift $b+2au$ and running cost $\langle u,au\rangle-f(x)$. The parameter $\lambda^{\!*}(f)$ can be thought of as the optimal ergodic value; see Ichihara [16]. Note then that the twisted process defined above corresponds to the optimally controlled diffusion. We refer to Ichihara [17, 16] and Kaise and Sheu [9] for some important results in this direction. For a potential $f$ that vanishes at infinity, Ichihara [16, 18] considers the ergodic control problem in Eq. 1.5, with a more general Hamiltonian and under scaling of the potential. When $f$ is nonnegative, it is shown that the value of the ergodic problem with potential $\beta f$, $\beta\in\mathbb{R}$, equals the eigenvalue $\lambda^{\!*}(\beta f)$, and $\nabla\psi^{*}$ is the optimal control when the parameter $\beta$ exceeds a critical value $\beta_{c}$, while below that critical value a bifurcation occurs. Analogous are the results in [19] for viscous Hamilton–Jacobi equations with $a$ the identity matrix and a Hamiltonian which is a power of the gradient term. Most of the above results are obtained for bounded, and Lipschitz continuous $a$, $b$, and $f$. In Theorems 2.7 and 2.8 we extend these results to measurable $b$ and $f$, and possibly unbounded $a$ and $b$.

Optimality for the ergodic problem is shown in [16, 18] via the study of the optimal finite horizon problem (Cauchy parabolic problem). Inevitably, in doing so, optimality is shown in a certain class of controls. To overcome this limitation, we take a different approach to the ergodic control problem in Eq. 1.5. As well known, ergodic control problems can be cast as infinite dimensional linear programs [20, 21]. Consider a controlled diffusion, with the control taking values in a space $\mathbb{U}$ with extended generator $\mathcal{A}$, where the ‘action’ $u\in\mathbb{U}$ enters implicitly as a parameter in $\mathcal{A}$. Let $\mathscr{R}\colon\mathbb{U}\to\mathbb{R}$ denote the running cost. The primal problem then can be written

[TABLE]

Here $\mathcal{P}({\mathbb{R}^{d}}\times\mathbb{U})$ denotes the class of probability measures on the Borel $\sigma$-field of ${\mathbb{R}^{d}}\times\mathbb{U}$. Its elements are called ergodic occupation measures (see [20]). The dual problem takes the form

[TABLE]

where $\mathcal{D}(\mathcal{A})$ denotes the domain of $\mathcal{A}$. In other words the dual problem is a maximization over subsolutions of the Hamilton–Jacobi–Bellman (HJB) equation. For non-degenerate diffusions with a compact action space $\mathbb{U}$, under the hypothesis that $\mathscr{R}$ is near-monotone, or under uniform ergodicity conditions, it is well known that we have strong duality, i.e., $\alpha_{*}=\alpha$. To the best of our knowledge, this has not been established for problems with non-compact action spaces. In Theorem 2.9 we establish strong duality for the ergodic problem in Eq. 1.5. In this result, the coefficients $b$ and $f$ are bounded and measurable, and $a$ is bounded, Lipschitz, and uniformly elliptic. Moreover, we establish the unicity of the optimal ergodic occupation measure, and as a result of this, the uniqueness of the optimal stationary Markov control. The methodology is general enough that can be applied to various classes of ergodic control problems that are characterized by viscous HJB equations.

The results in [17, 9] are obtained for smooth coefficients ($C^{2,\alpha}$), and under an assumption of exponential ergodicity (see Eq. 3.12 below). We provide a sufficient condition in (H2) under which strict monotonicity of the principal eigenvalue holds. It is also shown that the exponential ergodicity condition of [17, 9] actually implies (H2); thus (H2) is weaker. Moreover, Eq. 3.12 cannot hold for bounded coefficients $a$ and $b$. See Remark 3.4 for details. In Theorem 3.3 we cite a sufficient condition under which strict monotonicity of $\lambda^{\!*}(f)$ holds even when $a$ and $b$ are bounded. Let us also remark that the method of proof [17, 9] utilizes the smoothness of the coefficients $a$, $b$ and $f$. This is because a gradient estimate (Bernstein method) is required, which is not available under weaker regularity. But this amount of regularity might not be available in many situations, for instance in models with a measurable drift which are often encountered in stochastic control problems. Let us also mention the unpublished work of Kaise and Sheu in [22] that contains some results similar to ours, in particular, similar to the results in Section 3 and the pinned multiplicative ergodic theorems. These results are also obtained under sufficient smoothness of the coefficients $a$, $b$, and $f$.

In Section 4 we apply the above mentioned results to study the infinite horizon risk-sensitive control problem. We refer the reader to [10] where the importance of these control problems is discussed. Unfortunately, the development of the infinite horizon risk-sensitive control problem for controlled diffusions has not been completely satisfactory, and the same applies to controlled Markov chain models. Most of the available results have been obtained under restrictive settings, and a full characterization such as uniqueness of the solution to the risk-sensitive HJB equation, and verification of optimality results is lacking. Let us give a quick overview of the existing literature on risk-sensitive control in the context of controlled diffusions which is relevant to our problem. Risk-sensitive control for models with a constant diffusion matrix and asymptotically flat drift is studied by Fleming and McEneaney in [23]. Another particular setting is considered by Nagai [24], where the action space is the whole Euclidean space, and the running cost has a specific structure. Menaldi and Robin have considered models with periodic data [25]. Under the assumption of a near-monotone cost, the infinite horizon risk-sensitive control problem is studied in [10, 26, 27], whereas Biswas in [28] has considered this problem under the assumption of exponential ergodicity. Differential games with risk-sensitive type costs have been studied by Basu and Ghosh [29], Biswas and Saha [30], and Ghosh et. al. [31]. All the above studies, have obtained existence of a pair $(V,\lambda^{\!*})$ that satisfies the risk-sensitive HJB equation, with $\lambda^{\!*}$ the optimal risk-sensitive value, and show that any minimizing selector of the HJB is an optimal control. The works in [24, 25] address the existence and uniqueness of a solution to the HJB equation, in their particular set up, but do not contain any verification of optimality results. Two main results that are missing from the existing literature, with the exception of [10], are (a) uniqueness of the solution to the HJB equation, and (b) verification for optimal control.

Following the ergodic control paradigm, we can identify two classes of models: (i) models with a near-monotone running cost and finite optimal value, and no other hypotheses on the dynamics, and (ii) models that enjoy a uniform exponential ergodicity. Near-monotone running cost models are studied in [24, 10, 26, 27]; however, only [10] obtains a full characterization without imposing a blanket ergodicity hypothesis. Studies for models in class (ii) can be found in [23, 29, 28, 31].

In this paper we study models in class (ii). The results developed in Sections 2 and 3 enable us to obtain a full characterization of the risk-sensitive control problem in Section 4. The main hypotheses are Assumptions 4.1 and 4.2. Another interesting result that we establish in Section 4 is the continuity of the controlled principal eigenvalue with respect to (relaxed) stationary Markov controls (see Theorem 4.3). This facilitates establishing the existence of an optimal stationary Markov control for risk-sensitive control problems under risk-sensitive type constraints. Let us also remark that this existence result is far from being obvious, since the controlled risk-sensitive value is lower-semicontinuous with respect to Markov controls and the equality $\lambda^{\!*}(f)=\mathscr{E}(f)$ is not true in general. Moreover, the usual technique of Lagrange multipliers does not work in this situation, because of the non-convex nature of the optimization criterion.

To summarize the main contributions of the paper, we have established several characterizations of the property of strict monotonicity of the principal eigenvalue, and extended several results in the literature on viscous HJB equations with potentials $f$ vanishing at infinity, and smooth data, to measurable potential and drift (Theorems 2.1, 2.2, 2.6, 2.7, 2.8, 2.9, 2.10 and 3.2). We have also studied a general class of risk-sensitive control problems under a uniform ergodicity hypothesis, and established the uniqueness of a solution to the HJB equation and verification of optimality results (Theorems 4.1 and 4.2). Equally interesting are the continuity results of the controlled principal eigenvalue with respect to stationary Markov controls (Theorems 4.3 and 4.5).

The paper is organized as follows. Subsection 1.1 states the assumptions on the coefficients of the operator $\mathscr{L}$, and Subsection 1.2 summarizes the notation used in the paper. The first three subsections of Section 2 contain the main results on the principal eigenvalue under minimal assumptions, while Subsection 2.4 is devoted to operators with potential $f$ which vanishes at infinity. Section 3 improves on the results of Section 2, under the assumption that Eq. 1.2 is exponentially ergodic. Section 4 is dedicated to the infinite horizon, risk-sensitive optimal control problem.

1.1 Assumptions on the model

The following assumptions on the coefficients of $\mathscr{L}$ are in effect throughout the paper unless otherwise mentioned.

• (A1)

Local Lipschitz continuity: The function \upsigma\;=\;\bigl{[}\upsigma^{ij}\bigr{]}\,\colon\,\mathbb{R}^{d}\to\mathbb{R}^{d\times d} is locally Lipschitz in $x$ with a Lipschitz constant $C_{R}>0$ depending on $R>0$. In other words, with $\lVert\upsigma\rVert\coloneqq\sqrt{\operatorname{trace}(\upsigma\upsigma^{\mathsf{T}})}$, we have

[TABLE]

We also assume that b\;=\;\bigl{[}b^{1},\dotsc,b^{d}\bigr{]}^{\mathsf{T}}\,\colon\,\mathbb{R}^{d}\to\mathbb{R}^{d} is locally bounded and measurable.

• (A2)

Affine growth condition: $b$ and $\upsigma$ satisfy a global growth condition of the form

[TABLE]

for some constant $C_{0}>0$.

• (A3)

Nondegeneracy: For each $R>0$, it holds that

[TABLE]

and for all $\xi=(\xi_{1},\dotsc,\xi_{d})^{\mathsf{T}}\in\mathbb{R}^{d}$, where, as defined earlier, $a=\frac{1}{2}\upsigma\upsigma^{\mathsf{T}}$.

Let us remark that the assumptions (A1)–(A3) are not optimal, and can be weakened in many situations. For instance, if $\upsigma$ is continuous and its weak derivative lies in $L^{2(d+1)}_{\mathrm{loc}}({\mathbb{R}^{d}})$, then Eq. 2.1 has a unique strong solution (see [32]). The results in this paper can be extended to this setup as well.

1.2 Notation

The standard Euclidean norm in $\mathbb{R}^{d}$ is denoted by $\lvert\,\cdot\,\rvert$, and $\langle\,\cdot\,,\cdot\,\rangle$ denotes the inner product. The set of nonnegative real numbers is denoted by $\mathbb{R}_{+}$, $\mathbb{N}$ stands for the set of natural numbers, and $\mathds{1}$ denotes the indicator function. Given two real numbers $a$ and $b$, the minimum (maximum) is denoted by $a\wedge b$ ($a\vee b$), respectively. The closure, boundary, and the complement of a set $A\subset{\mathbb{R}^{d}}$ are denoted by $\bar{A}$, $\partial{A}$, and $A^{c}$, respectively. We denote by $\uptau(A)$ the first exit time of the process $\{X_{t}\}$ from the set $A\subset\mathbb{R}^{d}$, defined by

[TABLE]

The open ball of radius $r$ in $\mathbb{R}^{d}$, centered at the origin, is denoted by $B_{r}$, and we let $\uptau_{r}\coloneqq\uptau(B_{r})$, and $\breve{\uptau}_{r}\coloneqq\uptau(B^{c}_{r})$.

The term domain in $\mathbb{R}^{d}$ refers to a nonempty, connected open subset of the Euclidean space $\mathbb{R}^{d}$. For a domain $D\subset\mathbb{R}^{d}$, the space $C^{k}(D)$ ($C^{\infty}(D)$), $k\geq 0$, refers to the class of all real-valued functions on $D$ whose partial derivatives up to order $k$ (of any order) exist and are continuous. Also, $C^{k}_{b}(D)$ ($C_{b}^{\infty}(D)$) is the class of functions whose partial derivatives up to order $k$ (of any order) are continuous and bounded in $D$, and $C_{\mathrm{c}}^{k}(D)$ denotes the subset of $C^{k}(D)$, $0\leq k\leq\infty$, consisting of functions that have compact support. In addition, $C_{\mathrm{o}}({\mathbb{R}^{d}})$ denotes the class of continuous functions on ${\mathbb{R}^{d}}$ that vanish at infinity. By $C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ and $C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ we denote the subsets of $C_{\mathrm{c}}({\mathbb{R}^{d}})$ and $C_{\mathrm{o}}({\mathbb{R}^{d}})$, respectively, consisting of all non-trivial nonnegative functions. We use the term non-trivial to refer to a function that is not a.e. equal to [math]. The space $L^{p}(D)$, $p\in[1,\infty)$, stands for the Banach space of (equivalence classes of) measurable functions $f$ satisfying $\int_{D}\lvert f(x)\rvert^{p}\,\mathrm{d}{x}<\infty$, and $L^{\infty}(D)$ is the Banach space of functions that are essentially bounded in $D$. The standard Sobolev space of functions on $D$ whose generalized derivatives up to order $k$ are in $L^{p}(D)$, equipped with its natural norm, is denoted by $\mathscr{W}^{k,p}(D)$, $k\geq 0$, $p\geq 1$. For a probability measure $\mu$ in $\mathcal{P}({\mathbb{R}^{d}})$ and a real-valued function $f$ which is integrable with respect to $\mu$ we use the notation

[TABLE]

In general, if $\mathcal{X}$ is a space of real-valued functions on $Q$, $\mathcal{X}_{\mathrm{loc}}$ consists of all functions $f$ such that $f\varphi\in\mathcal{X}$ for every $\varphi\in C_{\mathrm{c}}^{\infty}(Q)$. In this manner we obtain for example the space $\mathscr{W}_{\mathrm{loc}}^{2,p}(Q)$.

We often use Krylov’s extension of the Itô formula for functions in $\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ [33, p. 122], which we refer to as the Itô–Krylov formula.

2 General results

Let $(\Omega,\mathfrak{F},\{\mathfrak{F}_{t}\},\operatorname{\mathbb{P}})$ be a given filtered probability space with a complete, right continuous filtration $\{\mathfrak{F}_{t}\}$. Let $W$ be a standard Brownian motion adapted to $\{\mathfrak{F}_{t}\}$. Consider the stochastic differential equation

[TABLE]

The third term on the right hand side of Eq. 2.1 is an Itô stochastic integral. We say that a process $X=\{X_{t}(\omega)\}$ is a solution of Eq. 2.1, if it is $\mathfrak{F}_{t}$-adapted, continuous in $t$, defined for all $\omega\in\Omega$ and $t\in[0,\infty)$, and satisfies Eq. 2.1 for all $t\in[0,\infty)$ a.s. It is well known that under (A1)–(A3), there exists a unique solution of Eq. 2.1 [34, Theorem 2.2.4]. We let $\operatorname{\mathbb{E}}_{x}$ denote the expectation operator on the canonical space of the process with $X_{0}=x$, and $\operatorname{\mathbb{P}}_{x}$ the corresponding probability measure. Recall that $\uptau(D)$ denotes the first exit time of the process $X$ from a domain $D$. The process $X$ is said be recurrent if for any bounded domain $D$ we have $\operatorname{\mathbb{P}}_{x}(\uptau(D^{c})<\infty)=1$ for all $x\in\bar{D}^{c}$. Otherwise the process is called transient. A recurrent process is said to be positive recurrent if $\operatorname{\mathbb{E}}_{x}[\uptau(D^{c})]<\infty$ for all $x\in\bar{D}^{c}$. It is known that for a non-degenerate diffusion the property of recurrence (or positive recurrence) is independent of $D$ and $x$, i.e., if it holds for some domain $D$ and $x\in\bar{D}^{c}$, then it also holds for every domain $D$, and all points $x\in\bar{D}^{c}$ (see [34, Lemma 2.6.12 and Theorem 2.6.10]). We define the extended operator $\mathscr{L}\colon C^{2}(\mathbb{R}^{d})\mapsto L_{\mathrm{loc}}^{\infty}({\mathbb{R}^{d}})$ associated to Eq. 2.1 by

[TABLE]

Let $f\colon{\mathbb{R}^{d}}\to\mathbb{R}$ be a locally bounded, Borel measurable function, which is bounded from below in ${\mathbb{R}^{d}}$, i.e., $\inf_{\mathbb{R}^{d}}f>-\infty$. We refer to a function $f$ with these properties as a potential, and let $\mathscr{L}^{f}\coloneqq\mathscr{L}+f$.

2.1 Risk-sensitive value and Dirichlet eigenvalues

The following lemma summarizes some results from [3, 5, 4] on the eigenvalues of the Dirichlet problem for the operator $\mathscr{L}^{f}$. For simplicity, we state it for balls $B_{r}$, instead of more general domains.

Lemma 2.1

For each $r\in(0,\infty)$ there exists a unique pair (\widehat{\Psi}_{r},\hat{\lambda}_{r})\in\bigl{(}\mathscr{W}_{\mathrm{loc}}^{2,p}(B_{r})\cap C(\bar{B}_{r})\bigr{)}\times\mathbb{R}, for any $p\in[1,\infty)$, satisfying $\widehat{\Psi}_{r}>0$ on $B_{r}$, $\widehat{\Psi}_{r}=0$ on $\partial B_{r}$, and $\widehat{\Psi}_{r}(0)=1$, which solves

[TABLE]

with $\mathscr{L}$ as defined in Eq. 2.2. Moreover, $\hat{\lambda}_{r}$ has the following properties:

The map $r\mapsto\hat{\lambda}_{r}$ is continuous and strictly increasing. 2. 2.

In its dependence on the function $f$, $\hat{\lambda}_{r}$ is nondecreasing, convex, and Lipschitz continuous (with respect to the $L^{\infty}$ norm), with Lipschitz constant $1$. In addition, if $f\lneqq f^{\prime}$, then $\hat{\lambda}_{r}(f)<\hat{\lambda}_{r}(f^{\prime})$.

Proof 1

Existence and uniqueness of the solution follow by [4, Theorem 1.1] (see also [3]). Part (a) follows by [5, Theorem 1.10], and (iii)–(iv) of [5, Proposition 2.3], while part (b) follows by [3, Proposition 2.1]. \qed

We refer to $(\widehat{\Psi}_{r},\hat{\lambda}_{r})$ as the eigensolution of the Dirichlet problem, or the Dirichlet eigensolution of $\mathscr{L}^{f}$ on $B_{r}$. Correspondingly, $\hat{\lambda}_{r}$ and $\widehat{\Psi}_{r}$ are referred to as the Dirichlet eigenvalue and Dirichlet eigenfunction, respectively.

Lemma 2.1 (a) motivates the following definition.

Definition 2.1

Let $f$ be a potential. The principal eigenvalue $\lambda^{\!*}(f)$ on ${\mathbb{R}^{d}}$ of the operator $\mathscr{L}^{f}$ given in Eq. 1.1 is defined as $\lambda^{\!*}(f)\coloneqq\lim_{r\to\infty}\,\hat{\lambda}_{r}(f)$.

For a potential $f$ we also define

[TABLE]

We refer to $\mathscr{E}(f)$ as the risk-sensitive average of $f$. This quantity plays a key role in our analysis.

We also compare Definition 2.1 with the following definition of the principal eigenvalue, commonly used in the pde literature [5].

[TABLE]

The following hypothesis is enforced throughout Section 2 without further mention, and it is repeated only for emphasis.

• (H1)

$f$ is a potential, and $\lambda^{\!*}(f)$ is finite.

Lemma 2.2

The following hold

For any $r>0$, the Dirichlet eigensolutions $(\widehat{\Psi}_{n},\hat{\lambda}_{n})$ in Eq. 2.3 have the following stochastic representation

[TABLE]

for all large enough $n\in\mathbb{N}$. 2. 2.

It holds that $\lambda^{\!*}(f)=\hat{\Lambda}(f)$. 3. 3.

Let $\Psi^{*}$ be any limit point of the Dirichlet eigensolutions $(\widehat{\Psi}_{n},\hat{\lambda}_{n})$ as $n\to\infty$, and $\mathscr{B}$ be an open ball centered at [math] such that $\lambda^{\!*}(f-h)+\sup_{\mathscr{B}^{c}}|h|<\lambda^{\!*}(f)<\infty$ for some bounded function $h$. Then with $\breve{\uptau}$ denoting the first hitting time of $\mathscr{B}$ we have

[TABLE]

Proof 2

Part (i) follows from [10, Lemma 2.10 (i)].

Turning to part (ii), suppose that $\lambda^{\!*}(f)$ is finite. Then it is standard to show that there exists a positive $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ which satisfies

[TABLE]

See [10, 26] for instance. It is then clear that $\lambda^{\!*}(f)\geq\hat{\Lambda}(f)$.

To show the converse inequality, suppose that a pair $(\varphi,\lambda)\in\mathscr{W}_{\mathrm{loc}}^{2,d}\times\mathbb{R}$, with $\varphi>0$, satisfies

[TABLE]

We claim that $\lambda^{\!*}(f)\leq\lambda$. If not, then we can find a pair $(\widehat{\Psi}_{r},\hat{\lambda}_{r})$ as in by Lemma 2.1, satisfying Eq. 2.3 and $\hat{\lambda}_{r}>\lambda$. By the Itô–Krylov formula [33, p. 122] we have

[TABLE]

Since $\varphi$ is positive, Eqs. 2.6 and 2.10 imply that we can scale it by multiplying with a constant $\kappa>0$ so that $\kappa\varphi-\widehat{\Psi}_{r}$ attains it minimum in $\bar{B}_{r}$ and this minimum value is [math]. Combining Eqs. 2.3 and 2.9, we obtain

[TABLE]

It then follows by the strong maximum principle [35, Theorem 9.6] that $\kappa\varphi-\widehat{\Psi}_{r}=0$ in $\bar{B}_{r}$, which is not possible since $\varphi>0$ on ${\mathbb{R}^{d}}$. This proves the claim. Since $\lambda$ was arbitrary, this implies that $\hat{\Lambda}(f)\geq\lambda^{\!*}(f)$, and thus we have equality.

It remains to prove Eq. 2.7. We follow the same argument as in [10, Lemma 2.10]. We fix $\mathscr{B}=B_{r}$. Letting $n\to\infty$ in Eq. 2.6 and applying Fatou’s lemma we obtain

[TABLE]

Thus, with $\tilde{\Psi}^{*}$ denoting a solution of Eq. 2.8, with $f$ replaced by $f-h$ and $\lambda=\lambda^{\!*}(f-h)$, we also have

[TABLE]

which implies that

[TABLE]

since $\tilde{\Psi}^{*}>0$ in ${\mathbb{R}^{d}}$. We write Eq. 2.6 as

[TABLE]

Note that since $\hat{\lambda}_{n}\nearrow\lambda^{\!*}(f)$, the first term on the right hand side of Eq. 2.13 is finite by Eq. 2.12 for all large enough $n$. Let

[TABLE]

The second term on the right hand side of Eq. 2.13 has the bound

[TABLE]

By the convergence of $\widehat{\Psi}_{n}\to\Psi^{*}$ as $n\to\infty$, uniformly on compact sets, and since $\widehat{\Psi}_{n}$ is bounded away from [math] in $\mathscr{B}$, uniformly in $n\in\mathbb{N}$, by Harnack’s inequality, we have $\kappa_{n}\to 0$ as $n\to\infty$. Therefore, the second term on the right hand side of Eq. 2.13 vanishes as $n\to\infty$. Also, since $\hat{\lambda}_{n}$ is nondecreasing in $n$, and $\hat{\lambda}_{n}\nearrow\lambda^{\!*}(f)$, we obtain

[TABLE]

by Eq. 2.12 and dominated convergence. Thus taking limits in Eq. 2.13 as $n\to\infty$, and using Eqs. 2.14 and 2.11, we obtain Eq. 2.7. This completes the proof. \qed

Combining Lemma 2.2 (ii) and [5, Theorem 1.4] we have the following result.

Corollary 2.1

There exists a positive $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p\geq 1$, satisfying

[TABLE]

if and only if $\lambda\geq\lambda^{\!*}(f)$.

As also mentioned in the introduction, throughout the rest of the paper, by an eigenpair $(\Psi,\lambda)$ of $\mathscr{L}^{f}$ we mean a positive function $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ and a scalar $\lambda\in\mathbb{R}$ that satisfy Eq. 2.15. In addition, the eigenfunction $\Psi$ is assumed to be normalized as $\Psi(0)=1$, unless indicated otherwise. When $\lambda$ is the principal eigenvalue, we refer to $(\Psi,\lambda)$ as a principal eigenpair. Note, that in view of the assumptions on the coefficients, any $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ which satisfies Eq. 2.15 belongs to $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, for all $p\in[1,\infty)$. Therefore, in the interest of notational economy, we refrain from mentioning the function space of solutions $\Psi$ of equations of the form Eq. 2.15, and any such solution is meant to be in $\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$. Moreover, since these are always strong solutions, we often suppress the qualifier ‘a.e.’, and unless a different domain is specified, such equations or inequalities are meant to hold on ${\mathbb{R}^{d}}$.

2.2 Summary of results

A major objective in this paper is to relate the properties of the eigenvalues $\lambda$ in Eq. 2.15 to the recurrence properties of the twisted process which is defined as follows. For an eigenfunction $\Psi$ satisfying Eq. 2.15 we let $\psi\coloneqq\log\Psi$. Then we can write Eq. 2.15 as

[TABLE]

The twisted process corresponding to an eigenpair $(\Psi,\lambda)$ of $\mathscr{L}^{f}$ is defined by the SDE

[TABLE]

Since $\psi\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p>d$, it follows that $\nabla\psi$ is locally bounded (in fact it is locally Hölder continuous), and therefore Eq. 2.17 has a unique strong solution up to its explosion time. We let $\widetilde{\mathscr{L}}^{\psi}_{\phantom{u}}$ denote the extended generator of Eq. 2.17, and $\widetilde{\operatorname{\mathbb{E}}}^{\psi}_{x}$ the associated expectation operator. The reader might have observed that the twisted process corresponds to Doob’s $h$-transformation of the operator $\mathscr{L}^{(f-\lambda)}$ with $h=\Psi$.

With $\Psi^{*}$ denoting a principal eigenfunction, i.e., an eigenfunction associated with $\lambda^{\!*}(f)$, we let $\psi^{*}\coloneqq\log\Psi^{*}$, and denote by $Y^{*}$ the corresponding twisted process. A twisted process corresponding to a principal eigenpair is called a ground state process, and the eigenfunction $\Psi^{*}$ is called a ground state.

Recall that $C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ denotes the collection of all non-trivial, nonnegative, continuous functions which vanish at infinity. We consider the following two properties of $\lambda^{\!*}(f)$.

• (P1)

Strict monotonicity at $f$.  For some $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ we have $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$.

• (P2)

Strict monotonicity at $f$ on the right.  For all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ we have $\lambda^{\!*}(f)<\lambda^{\!*}(f+h)$.

It follows by the convexity of $f\mapsto\lambda^{\!*}(f)$ that (P1) implies (P2).

Later, in Section 3, we provide sufficient conditions under which (P1) holds. Also, the finiteness of $\lambda^{\!*}(f)$ and $\lambda^{\!*}(f-h)$ is implicit in (P1). Indeed, since for every positive $\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, and $\lambda\in\mathbb{R}$ we have

[TABLE]

it follows that $\lambda^{\!*}(f-h)$ and $\lambda^{\!*}(f)$ are either both finite, or both equal to $\pm\infty$. It is also clear that $\lambda^{\!*}(f-h)\leq\lambda^{\!*}(f)$ always hold. As shown in Theorem 2.2, (P1) implies that $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$.

We introduce the following definition of exponential ergodicity which we often use.

Definition 2.2** (exponential ergodicity)**

The process $X$ governed by Eq. 1.2 is said to be exponentially ergodic if for some compact set $\mathscr{B}$ and $\delta>0$ we have \operatorname{\mathbb{E}}_{x}\bigl{[}\mathrm{e}^{\delta\,\uptau(\mathscr{B}^{c})}\bigr{]}<\infty, for all $x\in\mathscr{B}^{c}$.

The main results of this section center around the following theorem.

Theorem 2.1

Under (H1), the following hold:

A ground state process is recurrent if and only if $\lambda^{\!*}(f)$ is strictly monotone at $f$ on the right, in which case the principal eigenvalue $\lambda^{\!*}(f)$ is also simple, and the ground state $\Psi^{*}$ satisfies

[TABLE] 2. 2.

The ground state process is exponentially ergodic if and only if $\lambda^{\!*}(f)$ is strictly monotone at $f$. 3. 3.

If $\lambda>\lambda^{\!*}(f)$, the twisted process Eq. 2.17 corresponding to any solution $\psi$ of Eq. 2.16 is transient.

Proof 3

Part (a) follows by Lemmas 2.7, 2.3 and 2.3. Part (b) is the statement of Theorem 2.2, while part (c) is shown in Lemma 2.6. \qed

Theorem 2.1 should be compared with the results in [17, Theorem 2.2] and [9, Theorem 3.2 and 3.7]. The results in [17, 9] are obtained under a stronger hypothesis (same as Eq. 3.12 below) and for sufficiently regular coefficients. For a similar result in a bounded domain we refer the reader to [2, Theorem 4.2.4], where results are obtained for a certain class of operators with regular coefficients.

We remark that (P1) does not imply that the underlying process in Eq. 2.1 is recurrent. Indeed consider a one-dimensional diffusion with $b(x)=\frac{3}{2}x$ and $\upsigma=1$, and let $f(x)=x^{2}$. Then Eq. 2.15 holds with $\Psi(x)=\mathrm{e}^{-x^{2}}$ and $\lambda=-1$. But $b(x)+2a\nabla{\psi}=-\frac{1}{2}x$, so the twisted process is exponentially ergodic, while the original diffusion is transient.

The proof of Theorem 2.1 is divided in several lemmas which also contain results of independent interest. These occupy the next section.

2.3 Proof of Theorem 2.1 and other results

In the sequel, we often use the following finite time representation. This also appears in [10, Lemma 2.4] but in a slightly different form. Let $\uptau_{\infty}\coloneqq\lim_{n\to\infty}\,\uptau_{n}$ where $\uptau_{n}$ denotes the exit time from the ball $B_{n}$. Recall that if $(\Psi,\lambda)$ is an eigenpair of $\mathscr{L}^{f}$, and $\psi=\log\Psi$, then $\widetilde{\operatorname{\mathbb{E}}}_{x}^{\psi}$ denotes the expectation operator associated with the twisted process $Y$ in Eq. 2.17.

Lemma 2.3

If $(\Psi,\lambda)$ is an eigenpair of $\mathscr{L}^{f}$, then

[TABLE]

and for any function $g\in C_{\mathrm{c}}({\mathbb{R}^{d}})$, where $Y$ is the corresponding twisted process defined by Eq. 2.17.

Proof 4

The equation in Eq. 2.19 can be obtained by applying the Cameron–Martin–Girsanov theorem [36, p. 225]. Since $\psi$ and $f$ are not bounded, we need to localize the martingale. We use the first exit times $\uptau_{n}$ from $B_{n}$ as localization times. It is well-known that assumption (A2) implies that $\uptau_{n}\to\infty$ as $n\to\infty$ $\operatorname{\mathbb{P}}_{x}$-a.s. Applying the Itô–Krylov formula and using Eq. 2.16, we obtain

[TABLE]

Let $g$ be any nonnegative, continuous function with compact support. Then from 4 we obtain

[TABLE]

where in the last line we use Girsanov’s theorem. Given any bounded ball $\mathscr{B}$, by Itô’s formula and Fatou’s lemma, we obtain from Eq. 2.15 that

[TABLE]

Therefore, if we write

[TABLE]

we deduce that the first term on the right hand side is equal to [math] for all $n$ sufficiently large since $g$ is compactly supported, while the second term converges as $n\to\infty$ to the right hand side of Eq. 2.19 by Eq. 2.22 and dominated convergence. In addition, since $g$ has compact support, the term inside the expectation in the right hand side of 4 is bounded uniformly in $n$. Since also \widetilde{\operatorname{\mathbb{E}}}_{x}^{\psi}\bigl{[}g(Y_{\uptau_{n}})\,\Psi^{-1}(Y_{\uptau_{n}})\bigr{]}=0 for all sufficiently large $n$, letting $n\to\infty$ in 4, we obtain

[TABLE]

This proves Eq. 2.19. \qed

Recall that $\uptau_{\infty}\coloneqq\lim_{n\to\infty}\,\uptau_{n}$. An immediate corollary to Lemma 2.3 is the following.

Corollary 2.2

With $(\Psi,\lambda)$ as in Lemma 2.3, we have

[TABLE]

Proof 5

Choose a sequence of cut-off functions $g_{n}$ that approximates unity from below. Then Eq. 2.19 holds with $g$ replaced by $g_{n}\Psi$. Thus the result follows by letting $n\to\infty$ and applying the monotone convergence theorem. \qed

We are now ready to prove uniqueness of the principal eigenfunction.

Lemma 2.4

Under (P1) there exists a unique ground state $\Psi^{*}$ for $\mathscr{L}^{f}$, i.e., a positive $\Psi^{*}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, $\Psi^{*}(0)=1$, which solves

[TABLE]

Proof 6

Let $\Psi^{*}$ be a solution of Eq. 2.23 obtained as a limit of $\widehat{\Psi}_{r}$ (see Lemma 2.2). Thus by Lemma 2.2 (iii) we can find an open ball $\mathscr{B}$ such that

[TABLE]

with $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. Suppose that $\tilde{\Psi}$ is another principal eigenfunction of $\mathscr{L}^{f}$. By the Itô–Krylov formula and Fatou’s lemma, and since $\Psi^{*}$ is positive on $\bar{\mathscr{B}}$, we obtain

[TABLE]

It is clear by 6 that if $\tilde{\Psi}>\Psi^{*}$ on $\bar{\mathscr{B}}$, then $\tilde{\Psi}-\Psi^{*}>0$ on ${\mathbb{R}^{d}}$. Therefore, we can scale $\Psi^{*}$ by multiplying it with $\min_{\bar{\mathscr{B}}}\tfrac{\tilde{\Psi}}{\Psi^{*}}$ so that $\tilde{\Psi}$ touches $\Psi^{*}$ from above in $\bar{\mathscr{B}}$ at the points $\operatorname*{arg\,min}_{\bar{\mathscr{B}}}\tfrac{\tilde{\Psi}}{\Psi^{*}}$. Denoting this scaled $\Psi^{*}$ also as $\Psi^{*}$, it follows from 6 that $\tilde{\Psi}-\Psi^{*}$ is nonnegative in ${\mathbb{R}^{d}}$, and its minimum is [math] and attained in $\bar{\mathscr{B}}$. On the other hand, we have

[TABLE]

Thus $\tilde{\Psi}-\Psi^{*}=0$ by the strong maximum principle [35, Theorem 9.6], and this proves the result. \qed

We next show that (P1) implies the exponential ergodicity of $Y^{*}$.

Lemma 2.5

Assume (P1). Let $\Psi^{*}$ be the ground state of $\mathscr{L}^{f}$, and $\psi^{*}=\log\Psi^{*}$. Then the ground state process $Y^{*}$ governed by

[TABLE]

is exponentially ergodic. In particular, $Y^{*}$ is positive recurrent.

Proof 7

We first show that the finite time representation of $\Psi^{*}$ holds. Let $\tilde{\lambda}^{\!*}\coloneqq\lambda^{\!*}(f-h)$, and $\mathscr{B}$ be a ball as in Lemma 2.2 (iii). Recall that $\hat{\lambda}_{n}\to\lambda^{\!*}(f)$ as $n\to\infty$, and therefore, we have $\hat{\lambda}_{n}>\tilde{\lambda}^{\!*}+\sup_{\mathscr{B}^{c}}\lvert h\rvert$ for all sufficiently large $n$. Consider the following equations

[TABLE]

Choose $n$ large enough so that $\mathscr{B}\subset B_{n}$. We can scale $\tilde{\Psi}^{*}$, by multiplying it with a positive constant, so that $\tilde{\Psi}^{*}$ touches $\widehat{\Psi}_{n}$ from above. Next we show that it can only touch $\widehat{\Psi}_{n}$ in $\mathscr{B}$. Note that in $B_{n}\setminus\mathscr{B}$ we have

[TABLE]

Therefore, by the strong maximum principle, if $(\tilde{\Psi}^{*}-\widehat{\Psi}_{n})$ attains its minimum in $B_{n}\setminus\mathscr{B}$, then $(\tilde{\Psi}^{*}-\widehat{\Psi}_{n})=0$ in $B_{n}$, which is not possible. Thus $\tilde{\Psi}^{*}$ touches $\widehat{\Psi}_{n}$ in $\mathscr{B}$. Thus, applying Harnack’s inequality we can find a constant $\kappa_{1}$ such that $\kappa_{1}\tilde{\Psi}^{*}\geq\widehat{\Psi}_{n}$ for all sufficiently large $n$. On the other hand, by the Itô–Krylov formula and Fatou’s lemma we know that

[TABLE]

Applying the Itô–Krylov formula to Eq. 2.3 we have

[TABLE]

and letting $n\to\infty$, using Eq. 2.26 and the dominated convergence theorem, we obtain

[TABLE]

where $\Psi^{*}$ is the unique solution of Eq. 2.23. This proves the finite time representation. Thus it follows from Corollary 2.2 that Eq. 2.25 is regular, i.e., $\widetilde{\operatorname{\mathbb{P}}}_{x}^{\psi^{*}}(\uptau_{\infty}<\infty)=0$.

If we define $\Phi\coloneqq\frac{\tilde{\Psi}^{*}}{\Psi^{*}}$, a straightforward calculation shows that

[TABLE]

for some positive constants $C$ and $\epsilon$. Recall $\mathscr{B}$ from Lemma 2.2 (iii). It is easy to see from Eq. 2.7 that

[TABLE]

Thus $\Phi$ is uniformly bounded from below by a positive constant. Since $Y^{*}$ in Eq. 2.25 is regular, the Foster–Lyapunov inequality in Eq. 2.27 implies that $Y^{*}$ is exponentially ergodic. \qed

We denote the invariant measure of Eq. 2.25 by $\mu^{*}$. The following lemma shows that the twisted process is transient for any $\lambda>\lambda^{\!*}(f)$.

Lemma 2.6

Let $\Psi$ be an eigenfunction of $\mathscr{L}^{f}$ for an eigenvalue $\lambda>\lambda^{\!*}(f)$. Then the corresponding twisted process $Y$ is transient.

Proof 8

Let $\psi=\log\Psi$. If $\widetilde{\operatorname{\mathbb{P}}}_{x}^{\psi}(\uptau_{\infty}<\infty)>0$, then there is nothing to prove. So we assume the contrary. Hence from Lemma 2.3 we have

[TABLE]

for any continuous $g$ with compact support. Let $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$. By the Itô–Krylov formula and Fatou’s lemma, we have

[TABLE]

Thus, for $\delta=\lambda-\lambda^{\!*}(f)>0$, we obtain

[TABLE]

Combining Eqs. 2.28 and 2.29, we have

[TABLE]

Therefore, $Y$ is transient. \qed

Theorem 2.2

The following are equivalent.

The process $Y^{*}$, defined in Eq. 2.25, corresponding to some principal eigenpair \bigl{(}\Psi^{*},\lambda^{\!*}(f)\bigr{)} is exponentially ergodic. 2. 2.

It holds that $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$. 3. 3.

It holds that $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$ for some $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$.

Proof 9

(iii)$\,\Rightarrow\,$(i) follows from Lemma 2.5, and (ii)$\,\Rightarrow\,$(iii) is obvious.

We show that (i)$\,\Rightarrow\,$(ii). If $Y^{*}$ is exponentially ergodic, then there exists a ball $\mathscr{B}$ and $\delta>0$ such that

[TABLE]

Mimicking the calculations in the proof of Lemma 2.3, we obtain that

[TABLE]

for $g\in C_{\mathrm{c}}({\mathbb{R}^{d}})$. We apply this equation to an increasing sequence $\{g_{m}\}\subset C_{\mathrm{c}}({\mathbb{R}^{d}})$ which converges to $1$, and let first $m\to\infty$, and then $T\to\infty$, using Fatou’s lemma and the exponential ergodicity of $Y^{*}$, to obtain

[TABLE]

Let $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$. Since $h$ is bounded, it is easy to see that $\lambda^{\!*}(f-h)$ is finite. Let $\tilde{f}\coloneqq f-h$, and \bigl{(}\tilde{\Psi}^{*},\lambda^{\!*}(\tilde{f})\bigr{)} be a solution of

[TABLE]

which is obtained as a limit of Dirichlet eigensolutions as in Lemma 2.2. If $\lambda^{\!*}(\tilde{f})=\lambda^{\!*}(f)$, then in view of Eq. 2.30 and the calculations in the proof of Lemma 2.2 (iii), we have

[TABLE]

Applying the Itô–Krylov formula and Fatou’s lemma to Eq. 2.23, we obtain

[TABLE]

It follows by Eqs. 2.32 and 2.33 that we can multiply $\Psi^{*}$ with a suitable positive constant so that $\Psi^{*}-\tilde{\Psi}^{*}$ attains a minimum of [math] in $\mathscr{B}$. On the other hand, from Eqs. 2.23 and 2.31 we have

[TABLE]

Thus by strong maximum principle we have $\Psi^{*}=\tilde{\Psi}^{*}$. This, in turn, implies that $h\,\tilde{\Psi}^{*}=0$ by Eq. 2.34. But this is not possible. Hence we have $\lambda^{\!*}(\tilde{f})<\lambda^{\!*}(f)$, and the proof is complete. \qed

We define the Green’s measure $G_{\lambda}$, $\lambda\in\mathbb{R}$, by

[TABLE]

The density of the Green’s measure with respect to the Lebesgue measure is called the Green’s function. Existence of a Green’s function (and Green’s measure) is used by Pinsky [2, Chapter 4.3] in his definition of the generalized principal eigenvalue of $\mathscr{L}^{f}$. A number $\lambda\in\mathbb{R}$ is said to be subcritical if $G_{\lambda}$ possesses a density, critical if it is not subcritical and $\mathscr{L}^{f-\lambda}V=0$ has a positive solution $V$, and supercritical if it is neither subcritical nor critical.

The lemma which follows is an extension of [2, Theorem 4.3.4] where, under a regularity assumption on the coefficients, it is shown that a critical eigenvalue $\lambda$ is always simple. This result establishes several equivalences of the notion of criticality of $\lambda$.

Lemma 2.7

The following are equivalent.

The twisted process $Y$ corresponding to the eigenpair $(\Psi,\lambda)$ is recurrent. 2. 2.

$G_{\lambda}(g)$* is infinite for some $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$.* 3. 3.

For some open ball $\mathscr{B}$, and with $\breve{\uptau}=\breve{\uptau}(\mathscr{B})$, we have

[TABLE]

where $\Psi$ is an eigenfunction corresponding to the eigenvalue $\lambda$.

In addition, in (ii)–(iii)* “some” may be replaced by “all”, and if any one of (i)–(iii) holds, then $\lambda$ is a simple eigenvalue.*

Proof 10

The argument of this proof is inspired from [10, Theorem 2.8]. By Corollary 2.1 we have $\lambda\geq\lambda^{\!*}(f)$. Assume that (i) holds for some $\lambda\geq\lambda^{\!*}(f)$. Let $(\Psi,\lambda)$ be an eigenpair of $\mathscr{L}^{f}$. Then for any $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ we have from Lemma 2.3 that

[TABLE]

On the other hand, if $Y$ is recurrent, then

[TABLE]

Combining this with Eq. 2.35 we have $G_{\lambda}(g)=\infty$. Hence (ii) follows.

Next suppose that (ii) holds, i.e., $G_{\lambda}(g)=\infty$ for some $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ and $\lambda\geq\lambda^{\!*}(f)$. Applying the Itô–Krylov formula to $\mathscr{L}\Psi+(f-\lambda)\Psi=0$, we have

[TABLE]

for all $t\geq 0$, and for any bounded ball $\mathscr{B}$. Define $F_{\alpha}(x)\coloneqq f(x)-\lambda-\alpha$, and

[TABLE]

for $\alpha>0$, and some $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$. From Eq. 2.36 we have $\Gamma_{\alpha}<\infty$ for all $\alpha>0$. Moreover, $\Gamma_{\alpha}\to\infty$ as $\alpha\searrow 0$ by hypothesis. Choose $n_{0}$ large enough so that $\operatorname*{support}(g)\subset B_{n_{0}}$. Following [10, Theorem 2.8] we consider the positive solution $\varphi_{\alpha,n}\in\mathscr{W}_{\mathrm{loc}}^{2,p}(B_{n})\cap C(\bar{B}_{n})$ of

[TABLE]

for $n\geq n_{0}$. Since for every fixed $n$ we have

[TABLE]

by Eq. 2.36, applying the Itô–Krylov formula to Eq. 2.37, we obtain by [10, Theorem 2.8] that

[TABLE]

Since $\Gamma^{-1}_{\alpha}$ is bounded uniformly on $\alpha\in(0,1)$ by hypothesis, we can apply Harnack’s inequality for a class of superharmonic functions [37, Corollary 2.2] to conclude that $\{\varphi_{\alpha,n}\,,n\in\mathbb{N}\}$ is locally bounded, and therefore also uniformly bounded in $\mathscr{W}_{\mathrm{loc}}^{2,p}(B_{R})$, $p>d$, for any $R>0$. Thus, we have that $\varphi_{\alpha,n}\to\varphi_{\alpha}$ weakly in $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$ along some subsequence, and that $\varphi_{\alpha}$ satisfies

[TABLE]

by Eq. 2.37. Let $\mathscr{B}$ be an open ball centered at [math] such that $\operatorname*{support}(g)\subset\mathscr{B}$. Applying the Itô–Krylov formula to Eq. 2.37 we obtain

[TABLE]

with $\breve{\uptau}=\breve{\uptau}(\mathscr{B}^{c})$. As in the derivation of Eq. 2.38, using Eq. 2.36 and a similar argument we obtain

[TABLE]

Letting $n\to\infty$ along some subsequence, and arguing as above, we obtain a function $\varphi_{\alpha}$ which satisfies Eq. 2.39 and

[TABLE]

where Eq. 2.41 follows from Eq. 2.40. From Eq. 2.38 we have $\varphi_{\alpha}(0)=1$ for all $\alpha\in(0,1)$. Now applying Harnack’s inequality once again and letting $\alpha\searrow 0$, we deduce that $\varphi_{\alpha}$ converges weakly in $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p>d$, to some positive function $\Psi$ which satisfies $\mathscr{L}\Psi+F_{0}\,\Psi=0$ in ${\mathbb{R}^{d}}$, and

[TABLE]

This implies (iii).

Lastly, suppose that (iii) holds. In other words, there exists an eigenpair $(\Psi,\lambda)$ and an open ball $\mathscr{B}$ such that

[TABLE]

We first show that $\lambda$ is a simple eigenvalue, which implies that there is a unique twisted process $Y$ corresponding to $\lambda$. To establish the simplicity of $\lambda$ consider another eigenpair $(\tilde{\Psi},\lambda)$ of $\mathscr{L}^{f}$. By the Itô–Krylov formula we obtain

[TABLE]

Thus using Eq. 2.42 and an argument similar to Lemma 2.4 we can show that $\Psi=\tilde{\Psi}$. Then (iii)$\,\Rightarrow\,$(i) follows from [10, Lemma 2.6].

Uniqueness of the eigenfunction $\Psi$ follows from the stochastic representation in Eq. 2.42 and the proof of (iii)$\,\Rightarrow\,$(i). \qed

As an immediate corollary to Lemmas 2.6 and 2.7 we have the following.

Corollary 2.3

Let $(\Psi,\lambda)$ be an eigenpair of $\mathscr{L}^{f}$ which satisfies

[TABLE]

for some bounded open ball $\mathscr{B}$ in ${\mathbb{R}^{d}}$. Then $\lambda=\lambda^{\!*}(f)$, and it is a simple eigenvalue.

Theorem 2.3 below is a generalization of [2, Theorem 4.7.1] in ${\mathbb{R}^{d}}$, which is stated in bounded domains, and for bounded and smooth coefficients. It is shown in [2] that for smooth bounded domains, the Green’s measure is not defined at the critical value $\lambda^{\!*}$ [2, Theorem 3.2]. But by Theorem 2.3 below we see that this is not the case on ${\mathbb{R}^{d}}$. In fact, [2, Theorem 4.3.2] shows that $\lambda^{*}$ could be either subcritical or critical in the sense of Pinsky. We show that the criticality of $\lambda^{*}$ is equivalent to the strict monotonicity of $\lambda^{\!*}(f)$ on the right, i.e., $\lambda^{\!*}(f)<\lambda^{\!*}(f+h)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$.

Theorem 2.3

A ground state process is recurrent if and only if $\lambda^{\!*}(f)<\lambda^{\!*}(f+h)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$.

Proof 11

Suppose first that a ground state process corresponding to $\lambda^{\!*}(f)$ is recurrent. Then $G_{\lambda^{\!*}}(g)=\infty$ for all $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ by Lemma 2.7. Let $\tilde{f}=f+h$ and $\tilde{\lambda}^{\!*}\coloneqq\lambda^{\!*}(f+h)$. Suppose that $\lambda^{\!*}=\tilde{\lambda}^{\!*}$. Let $\tilde{\Psi}$ be a principal eigenfunction of $\mathscr{L}^{\tilde{f}}$, i.e.,

[TABLE]

Writing Eq. 2.43 as $\mathscr{L}\tilde{\Psi}+(f-\lambda^{\!*})\tilde{\Psi}=-h\tilde{\Psi}$, and applying the Itô–Krylov formula, followed by Fatou’s lemma, we obtain

[TABLE]

which contradicts the property that $G_{\lambda^{\!*}}(g)=\infty$ for all $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$. Therefore, $\lambda^{\!*}(f)<\lambda^{\!*}(f+h)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$.

To prove the converse, suppose that $Y^{*}$ is transient. Then for $g\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ with $B_{1}\subset\operatorname*{support}(g)$ we have $G_{\lambda^{\!*}}(g)<\infty$. Following the arguments in the proof of (ii)$\,\Rightarrow\,$(iii) in Lemma 2.7, we obtain a positive $\Phi$ satisfying

[TABLE]

Let $\varepsilon=\Gamma_{0}^{-1}\min_{B_{1}}\frac{g}{\Phi}$. Then from Eq. 2.44 we have

[TABLE]

This implies that $\lambda^{\!*}(f+\varepsilon\mathds{1}_{B_{1}})\leq\lambda^{\!*}(f)$ by Lemma 2.2 (ii). Thus $\lambda^{\!*}(f+\varepsilon\mathds{1}_{B_{1}})=\lambda^{\!*}(f)$. Therefore, if $\lambda^{\!*}(f)<\lambda^{\!*}(f+h)$ for all $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$, then $Y^{*}$ has to be recurrent. This completes the proof. \qed

It is well known that a (null) recurrent diffusion $\{X_{t}\}$ with locally uniformly elliptic and Lipschitz continuous $a$, and locally bounded measurable drift, admits a $\sigma$-finite invariant probability measure $\nu$ which is a Radon measure on the Borel $\sigma$-field of ${\mathbb{R}^{d}}$ [38]. This measure is equivalent to the Lebesgue measure and is unique up to a multiplicative constant. Theorem 8.1 in [38] states that if $g$ and $h$ are real-valued functions which are integrable with respect to the measure $\nu$ then

[TABLE]

Suppose $g\colon{\mathbb{R}^{d}}\to\mathbb{R}_{+}$ is a non-trivial function. Select $h$ as the indicator function of some open ball. Then it is well known that the expectation of $Y^{h}_{t}\coloneqq\int_{0}^{t}h(X_{t})\,\mathrm{d}{t}$ tends to $\infty$ as $t\to\infty$. Adopt the analogous notation $Y^{g}_{t}$, and let $\alpha=\frac{\nu(g)}{2\nu(h)}$. Let $M>0$ be arbitrary, and select $t_{0}$ large enough such that \operatorname{\mathbb{E}}\bigl{[}Y^{h}_{t_{0}}\bigr{]}\geq 2M. Then of course we may find a positive constant $\kappa$ such \operatorname{\mathbb{E}}\bigl{[}Y^{h}_{t_{0}}\,\mathds{1}_{\{Y^{h}_{t_{0}}\leq\kappa\}}\bigr{]}\geq M. Since $Y^{h}_{t}$ and $Y^{g}_{t}$ are nondecreasing in $t$, it follows by Eq. 2.45 that

[TABLE]

This of course implies, using dominated convergence, that \liminf_{t\to\infty}\,\operatorname{\mathbb{E}}\bigl{[}Y^{g}_{t}\,\mathds{1}_{\{Y^{h}_{t_{0}}\leq\kappa\}}\bigr{]}\geq\alpha M. Since $M$ was arbitrary, this shows that \operatorname{\mathbb{E}}\bigl{[}Y^{g}_{t}\bigr{]}\to\infty as $t\to\infty$, or equivalently that $\int_{0}^{\infty}\operatorname{\mathbb{E}}_{x}[g(X_{t})]\,\mathrm{d}{t}=\infty$. Using this property in the proof of Theorem 2.3 we obtain the following corollary.

Corollary 2.4

For $\lambda^{\!*}(f)$ to be strictly monotone at $f$ on the right it is sufficient that there exists some non-trivial Borel measurable bounded function $g\colon{\mathbb{R}^{d}}\to\mathbb{R}_{+}$ with compact support satisfying $\lambda^{\!*}(f+\epsilon\,g)>\lambda^{\!*}(f)$ for all $\epsilon>0$.

2.3.1 Minimal growth at infinity

We next discuss the property known as minimal growth at infinity [5, Definition 8.2]. As shown in [5, Proposition 8.4], minimal growth at infinity implies that the eigenspace corresponding to the eigenvalue $\lambda^{\!*}(f)$ is one dimensional, i.e., $\lambda^{\!*}(f)$ is simple. We start with the following definition, which is a variation of [5, Definition 8.2].

Definition 2.3

A positive function $\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ is said to be a solution of minimal growth at infinity of $\mathscr{L}^{f}\varphi-\lambda\varphi=0$, if for any $r>0$ and any positive function $v\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}}\setminus B_{r})$ satisfying $\mathscr{L}^{f}v-\lambda v\leq 0$ a.e., in $B_{r}^{c}$, there exists $R>r$ and $k>0$ such that $k\varphi\leq v$ in $B^{c}_{R}$.

Define the generalized principal eigenvalue of $\mathscr{L}^{f}$ in the domain $D$ by

[TABLE]

Note that $\lambda_{1}(f,{\mathbb{R}^{d}})=\hat{\Lambda}(f)=\lambda^{\!*}(f)$. It is also clear from this definition that for $D_{1}\subset D_{2}$ we have $\lambda_{1}(f,D_{1})\leq\lambda_{1}(f,D_{2})$.

It is shown in [5, Theorem 8.5] that the hypothesis

1. (A1)

$\lim_{r\to\infty}\;\lambda_{1}(f,B_{r}^{c})\;<\;\lambda^{\!*}(f)$

implies that the ground state $\Psi^{*}$ of $\mathscr{L}^{f}$ is a solution of minimal growth at infinity.

On the other hand, the following result has been established in [39, Theorem 2.1].

Theorem 2.4

The ground state $\Psi^{*}$ of $\mathscr{L}^{f}$ is a solution of minimal growth at infinity of $\mathscr{L}^{f}\Psi^{*}-\lambda^{\!*}(f)\Psi^{*}=0$ if and only if $\lambda^{\!*}(f)$ is strictly monotone at $f$ on the right.

It thus follows by the above results that (A1) is a sufficient condition for strict monotonicity of $\lambda^{\!*}(f)$ on the right. It turns out that (A1) is equivalent to strict monotonicity and, moreover, the map $r\mapsto\lambda_{1}(f,B_{r}^{c})-\lambda^{\!*}(f)$ is either negative on $(0,\infty)$ or identically equal to [math]. This is the subject of the following theorem.

Theorem 2.5

The following are equivalent.

$\exists\,r>0\,\colon\;\lambda_{1}(f,B_{r}^{c})<\lambda_{1}(f,{\mathbb{R}^{d}})$. 2. 2.

$\lambda^{\!*}(f)$* is strictly monotone at $f$.* 3. 3.

$\lambda_{1}(f,B_{r}^{c})<\lambda_{1}(f,{\mathbb{R}^{d}})\quad\forall\,r>0$.

Proof 12

It easily follows by Lemmas 2.5 and 2.2 and the definition of $\lambda_{1}$ that (b)$\,\Rightarrow\,$(c). Thus it remains to prove that (a)$\,\Rightarrow\,$(b). Suppose that $\lambda\equiv\lambda_{1}(f,B_{\bar{r}}^{c})<\lambda_{1}(f,{\mathbb{R}^{d}})=\lambda^{\!*}(f)$ for some $\bar{r}>0$. Using the Dirichlet eigenvalues for the annulus $\mathscr{B}_{r}\setminus\bar{B}_{\bar{r}}$, for $r>\bar{r}$, and letting $r\to\infty$, we can construct a solution $\psi\in\mathscr{W}_{\mathrm{loc}}^{2,d}(\bar{B}_{\bar{r}}^{c})$ of $\mathscr{L}\psi+f\psi=\lambda\psi$ on $B_{\bar{r}}^{c}$, with $\psi>0$ on $\bar{B}_{\bar{r}}^{c}$, and $\psi=0$ on $\partial B_{\bar{r}}$. Then $\psi$ is bounded away from [math] on $\partial B_{r^{\prime}}$ for all $r^{\prime}>r$. Using any $r^{\prime}>r$, we extend $\psi$ smoothly inside $B_{r^{\prime}}$ to obtain some function $\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ which is strictly positive on ${\mathbb{R}^{d}}$ and agrees with $\psi$ on $B_{r^{\prime}}^{c}$. Let $h\coloneqq\lambda\varphi-\mathscr{L}\varphi-f\varphi$, and $\tilde{f}\coloneqq f+\frac{h}{\varphi}$. Then $\mathscr{L}\varphi+\tilde{f}\varphi\;=\;\lambda\varphi$, and therefore, we have

[TABLE]

which implies strict monotonicity at $f$, and completes the proof. \qed

2.4 Potentials $f$ vanishing at infinity

Let $\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$ denote the class of bounded Borel measurable functions which are vanishing at infinity, i.e., satisfying $\lim_{R\to\infty}\;\sup_{B^{c}_{R}}\,\lvert f\rvert=0$, and $\mathcal{B}^{+}_{\mathrm{o}}({\mathbb{R}^{d}})$ the class of nonnegative functions in $\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$ which are not a.e. equal to [math].

Theorem 2.6 which follows is a (pinned) multiplicative ergodic theorem (compare with [22, Theorem 7.1]). Note that the continuity result in this theorem is stronger than that of [5, Proposition 9.2]. See also Remark 4.1 on the continuity of $\lambda^{\!*}(f)$ for a larger class of $f$. We introduce the eigenvalue $\lambda^{\prime\prime}(f)$ defined by

[TABLE]

Theorem 2.6

Let $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$. If the solution of Eq. 2.1 is recurrent, then $\lambda^{\!*}(f)=\lambda^{\prime\prime}(f)=\mathscr{E}(f)$. In addition, if the solution of Eq. 2.1 is positive recurrent with invariant measure $\mu$, and $\int_{{\mathbb{R}^{d}}}f\,\mathrm{d}{\mu}>0$, the following hold:

• (a)

for any measurable $g$ with compact support we have

[TABLE]

for some positive constant $C_{g}$. Moreover, the corresponding twisted process $Y^{*}$ is exponentially ergodic.

• (b)

If $f_{n}$ is a sequence of functions in $\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$ satisfying $\sup_{n}\lVert f_{n}\rVert_{\infty}<\infty$, and converging to $f$ in $L_{\mathrm{loc}}^{1}({\mathbb{R}^{d}})$, and also uniformly outside some compact set $K\subset{\mathbb{R}^{d}}$, then $\lambda^{\!*}(f_{n})\to\lambda^{\!*}(f)$.

Proof 13

Applying the Itô–Krylov formula to $\mathscr{L}\varphi+(f-\lambda)\varphi\leq 0$, it is easy to see that $\mathscr{E}(f)\leq\lambda^{\prime\prime}(f)$. Also, from [10, Lemma 2.3] we have $\lambda^{\!*}(f)\leq\mathscr{E}(f)$. Thus we obtain $\lambda^{\!*}(f)\leq\mathscr{E}(f)\leq\lambda^{\prime\prime}(f)$. If $\lambda^{\!*}(f)\geq\lim_{\lvert x\rvert\to\infty}\,f(x)$, then by [5, Theorem 1.9 (iii)] we have $\lambda^{\!*}(f)=\lambda^{\prime\prime}(f)$ which in turn implies that $\lambda^{\!*}(f)=\mathscr{E}(f)=\lambda^{\prime\prime}(f)$. On the other hand, if $\lambda^{\!*}(f)<\lim_{\lvert x\rvert\to\infty}\,f(x)$, then $f$ is near-monotone, relative to $\lambda^{\!*}(f)$, in the sense of [10]. Applying [10, Lemma 2.1] we again obtain $\lambda^{\!*}(f)=\mathscr{E}(f)=\lambda^{\prime\prime}(f)$.

We now turn to part (a). Applying Jensen’s inequality it is easy to see that $\mathscr{E}(f)\geq\int f\,\mathrm{d}{\mu}>0$. Therefore, $\lambda^{\!*}(f-f^{+})\leq 0<\lambda^{\!*}(f)$. Taking $h=f^{+}$ and mimicking the arguments of Theorem 2.1 we see that $Y^{*}$ is exponentially ergodic. Let $\mu^{*}$ be the unique invariant measure of $Y^{*}$. Then Eq. 2.47 follows from Eq. 2.28 and [40, Theorem 1.3.10] with $C_{g}\;=\;\int\frac{g}{\Psi^{*}}\,\mathrm{d}{\mu^{*}}$.

Next we prove part (b). By the first part of the theorem we have $\lambda^{\!*}(f_{n})=\mathscr{E}(f_{n})$ for all $n$, and by the lower-semicontinuity property of $\lambda^{\!*}$ it holds that $\liminf_{n\to\infty}\,\lambda^{\!*}(f_{n})\geq\lambda^{\!*}(f)$. Let $h\in C_{\mathrm{c}}^{+}({\mathbb{R}^{d}})$ and $\tilde{f}=f-h$. Then by Theorem 2.2 we have $2\delta:=\lambda^{\!*}(f)-\lambda^{\!*}(\tilde{f})>0$. Choose a open ball $\mathscr{B}$, containing $K$, such that $\sup_{x\in\mathscr{B}^{c}}|f_{n}-f|<\delta$ and $\lambda^{\!*}(f_{n})>\lambda^{\!*}(f)-\delta$ for all sufficiently large $n$. Let $(\Psi^{*}_{n},\lambda^{\!*}(f_{n}))$ denote the principal eigenpair. Then

[TABLE]

We can choose $\mathscr{B}$ large enough such that

[TABLE]

where $\breve{\uptau}=\breve{\uptau}(\mathscr{B})$. Suppose $\limsup_{n\to\infty}\lambda^{\!*}(f_{n})=\Lambda$. It is standard to show that for some positive $\Psi$, it holds that $\Psi^{*}_{n}\to\Psi$ weakly in $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p>d$, as $n\to\infty$, and therefore, from Eq. 2.48 we have

[TABLE]

Therefore, $\Lambda\geq\lambda^{\!*}(f)$. Note that on $\mathscr{B}^{c}$ we have

[TABLE]

for all $n$ sufficiently large. Since \operatorname{\mathbb{E}}_{x}\bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}[f(X_{t})-\lambda^{\!*}(\tilde{f})]\,\mathrm{d}{t}}\,\bigr{]}<\infty, passing to the limit in Eq. 2.49, and using the dominated convergence theorem, we obtain that

[TABLE]

Therefore, $\Lambda=\lambda^{\!*}(f)$ by Corollary 2.3. This completes the proof. \qed

We pause for a moment to provide an example where (P2) holds but (P1) fails.

Example 2.1

Let $d=2$ and $\mathscr{L}=\Delta$. If $f=0$, then the ground state is a constant function, and in turn, the ground state diffusion is a two dimensional Brownian motion, hence recurrent. It follows that $\lambda^{\!*}$ is strictly monotone on the right at [math]. Now let $f$ a non-trivial non-negative continuous function with compact support. It is clear that $\lambda^{\!*}(\beta f)\leq 0$ for $\beta\leq 0$. On the other hand, by Theorem 2.6, we have $\lambda^{\!*}(\beta f)=\mathscr{E}(\beta f)$ for all $\beta\in\mathbb{R}$. Therefore, for $\beta\leq 0$, we have

[TABLE]

Thus $\lambda^{\!*}(\beta f)=0$ for all $\beta\leq 0$, which implies that $\lambda^{\!*}$ it is not strictly monotone at [math].

In the rest of this section we show how the previous development can be used to obtain results analogous to those reported in [16], without imposing any smoothness assumptions on the coefficients. With $\breve{\psi}=-\log\Psi^{*}=-\psi^{*}$, we have

[TABLE]

Note that Eq. 2.50 is a particular form of a more general class of quasilinear pdes of the form

[TABLE]

where the function $H(x,p)$, with $(x,p)\in{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}$, serves as a Hamiltonian. Let $f$ be a non-constant, nonnegative continuous function satisfying $\lim_{\lvert x\rvert\to\infty}f(x)=0$, and define $\Lambda_{\beta}\coloneqq\lambda^{\!*}(\beta f)$, $\beta\in\mathbb{R}$. Then by [5, Proposition 2.3 (vii)] we know that $\beta\mapsto\Lambda_{\beta}$ is non-decreasing and convex. For the diffusion matrix $a$ equal the identity, Ichihara studies some qualitative properties of $\Lambda_{\beta}$ in [16] associated to the pde Eq. 2.51, and their relation to the recurrence and transience behavior of the process with generator

[TABLE]

It is clear that if $H(x,p)=-\langle b(x),p\rangle+\langle p,a(x)p\rangle$, then $\mathscr{A}^{\breve{\psi}}$ is the generator of the twisted process $Y^{*}$ corresponding to $\Psi^{*}$. One of the key assumptions in [16, Assumption (H1) (i)] is that $H(x,p)\geq H(x,0)=0$ for all $x$ and $p$. Note that this forces $b$ to be [math].

Let

[TABLE]

It is easy to see that $\beta_{c}\in[-\infty,\infty]$. The following result is an extension of [16, Theorems 2.2 and  2.3] to measurable drifts $b$ and potentials $f$.

Theorem 2.7

Let $f\in\mathcal{B}^{+}_{\mathrm{o}}({\mathbb{R}^{d}})$. Then the twisted process $Y^{*}=Y^{*}(\beta)$ corresponding to the eigenpair $(\Psi^{*}_{\beta},\Lambda_{\beta})$ is transient for $\beta<\beta_{c}$, exponentially ergodic for $\beta>\beta_{c}$, and, provided $f=0$ a.e. outside some compact set, it is recurrent for $\beta=\beta_{c}$. In addition, the following hold.

*If $\mathscr{L}^{0}$ is self-adjoint *(i.e., $\mathscr{L}^{0}=\partial_{i}(a^{ij}\partial_{j})$**), with the matrix $a$ bounded, uniformly elliptic and radially symmetric in ${\mathbb{R}^{d}}$, and the solution of Eq. 2.1 is transient, then $\beta_{c}\geq 0$. Also $\Lambda_{\beta}\geq 0$ for all $\beta\in\mathbb{R}$. 2. 2.

Provided that the solution of Eq. 2.1 is recurrent, then $\beta_{c}<0$ if it is exponentially ergodic, and $\beta_{c}=0$ otherwise. 3. 3.

Assume that $\beta>\beta_{c}$, and that Eq. 2.1 is recurrent in the case that $\Lambda_{\beta}\leq 0$. Let $\Psi^{*}_{\beta}$ and $\mu^{*}_{\beta}$ denote the ground state and the invariant probability measure of the ground state diffusion, respectively, corresponding to $\Lambda_{\beta}$. Then it holds that

[TABLE]

where, as usual, $\psi^{*}_{\beta}=\log\Psi^{*}_{\beta}$.

Proof 14

The first part of the proof follows from Theorems 2.1 and 2.3, and Corollary 2.4. Next we proceed to prove (i). Suppose $\beta_{c}<0$. Then $Y^{*}=Y^{*}(0)$, i.e., the twisted process corresponding to $\Lambda_{0}$, is exponentially ergodic. By [5, Theorem 1.9 (i)–(ii)] we have $\Lambda_{0}=\mathscr{E}(0)=0$. Moreover, $\Psi^{*}_{0}=1$ is a ground state. Therefore, the twisted process must be given by Eq. 2.1, which is transient by hypothesis. This is a contradiction. Hence $\beta_{c}\geq 0$. Since $\beta\mapsto\Lambda_{\beta}$ is convex, it follows that $\Lambda_{\beta}$ is constant in $(-\infty,\beta_{c}]\ni 0$. Hence $\Lambda_{\beta}=\Lambda_{0}=0$ for $\beta\leq\beta_{c}$. This proves (i).

We now turn to part (ii). By Theorem 2.6 we have $\Lambda_{\beta}=\mathscr{E}(\beta f)$. We claim that if the solution of Eq. 2.1 is recurrent then $\lambda^{*}(\beta f)>0$, whenever $\beta>0$. Indeed, arguing by contradiction, if $\lambda^{*}(\beta f)=0$ for some $\beta>0$, then $\mathscr{L}\Psi^{*}_{\beta}=-\beta f\Psi^{*}_{\beta}$ on ${\mathbb{R}^{d}}$, which implies that that $\Psi^{*}_{\beta}(X_{t})$ is a nonnegative supermartingale, and since it is integrable, it converges a.s. Since the process $X$ is recurrent, this implies that $\Psi^{*}_{\beta}$ must equal to a constant, which, in turn, necessitates that $f=0$, a contradiction. This proves the claim, which in turn implies that if the solution of Eq. 2.1 is recurrent then $\beta_{c}\leq 0$. Now suppose that $\beta_{c}$ is negative. Then the twisted process corresponding to $\beta=0$ is exponentially ergodic by Theorem 2.1. Since $\Psi^{*}_{0}=1$, the ground state diffusion for $\beta=0$ agrees with Eq. 2.1, which implies that the latter is exponentially ergodic.

Next, suppose that $X$, and therefore also $Y^{*}(0)$ is exponentially ergodic. It then follows from Theorem 2.2 that $\beta\mapsto\Lambda_{\beta}$ is strictly monotone at [math]. This of course implies that $\beta_{c}<0$. The proof of part (ii) is now complete.

Next we prove part (iii). We distinguish two cases.

Case 1.* Suppose $\Lambda_{\beta}>0$. Let {\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}={\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}_{\beta}\coloneqq(\Psi^{*}_{\beta})^{-1} and \breve{\psi}\coloneqq\log{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}. Then *

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}$

$\textstyle\Psi$

• satisfies*

[TABLE]

Since $\beta f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, there exists $\epsilon_{\circ}>0$ and a ball $\mathscr{B}$ such that $\beta f-\Lambda_{\beta}<-\epsilon_{\circ}$ for all $x\in\mathscr{B}^{c}$. Applying the Feynman–Kac formula, it follows from **[10*, Lemma 2.1]** that \inf_{\mathbb{R}^{d}}\,{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}=\min_{\bar{\mathscr{B}}}\,{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}. Thus *

$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}$

$\textstyle\Psi$

• is bounded away from [math] on ${\mathbb{R}^{d}}$. Let $Y^{*}$ denote the ground state process corresponding to the eigenvalue $\Lambda_{\beta}$. Simplifying the notation we let $\widetilde{\operatorname{\mathbb{E}}}_{x}^{*}\coloneqq\widetilde{\operatorname{\mathbb{E}}}_{x}^{\psi^{*}_{\beta}}$. By the exponential Foster–Lyapunov equation Eq. 2.53 we have that (see [34, Lemma 2.5.5])*

[TABLE]

Using this estimate together with the fact that \inf_{{\mathbb{R}^{d}}}{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}>0, we obtain

[TABLE]

Next, we show that

[TABLE]

where $\uptau_{R}$ denotes the exit time from the ball $B_{R}$. First, there exists some constant $k_{0}$ such that \bigl{(}\beta f-\Lambda_{\beta}\bigr{)}{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}\leq k_{0} on ${\mathbb{R}^{d}}$. Thus \widetilde{\operatorname{\mathbb{E}}}_{x}^{*}\bigl{[}{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}(Y^{*}_{t})\bigr{]}\leq k_{0}t+{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}(x) by Eq. 2.53, and of course also \widetilde{\operatorname{\mathbb{E}}}_{x}^{*}\bigl{[}{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}(Y^{*}_{t\wedge\uptau_{R}})\bigr{]}\leq k_{0}t+{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}(x) for all $R>0$. Let $\Gamma(R,m)\coloneqq\{x\in\partial B_{R}\colon\lvert\breve{\psi}(x)\rvert\geq m\}$ for $m\geq 1$. Then

[TABLE]

Taking limits as $R\to\infty$, and since $m\in\mathbb{R}_{+}$ is arbitrary, it follows that

[TABLE]

Write

[TABLE]

Without loss of generality we assume {\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}\geq 1. Since \lvert\breve{\psi}\rvert\leq{\mathchoice{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\displaystyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\displaystyle\Psi}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower 7.09259pt\hbox{\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule height=0.0pt,width=7.7778pt}}}}}\cr\hbox{\textstyle\Psi}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower 6.40926pt\hbox{\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule height=0.0pt,width=5.44446pt}}}}}\cr\hbox{\scriptstyle\Psi}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower 5.9537pt\hbox{\scriptscriptstyle\widehat{\vrule width=0.0pt,height=3.41666pt\vrule height=0.0pt,width=3.8889pt}}}}}\cr\hbox{\scriptscriptstyle\Psi}}}}}, an application of Fatou’s lemma shows that

[TABLE]

We use this together with Eqs. 2.57 and 2.58 to obtain Eq. 2.56.

We write Eq. 2.50 as

[TABLE]

Let $F\coloneqq\langle\nabla\breve{\psi},a\nabla\breve{\psi}\rangle-\beta f=\langle\nabla\psi^{*}_{\beta},a\nabla\psi^{*}_{\beta}\rangle-\beta f$. Applying the Itô–Krylov formula to 14, we obtain

[TABLE]

Letting $R\to\infty$ in Eq. 2.60, using Eq. 2.56, then dividing by $t$ and letting $t\to\infty$, using Eq. 2.55 and Birkhoff’s ergodic theorem, we obtain

[TABLE]

which is the assertion in part (iii).

Case 2.* Suppose $\Lambda_{\beta}\leq 0$ and Eq. 2.1 is recurrent. The case $\Lambda_{\beta}=0$ is then trivial, since $\nabla\psi^{*}_{0}=0$, so we assume that $\Lambda_{\beta}<0$. Then Eq. 2.1 is exponentially ergodic by part (ii), and thus $\Psi^{*}_{\beta}$ is bounded below in ${\mathbb{R}^{d}}$ by [10, Lemma 2.1]. With $\psi^{*}=\psi^{*}_{\beta}=\log\Psi^{*}_{\beta}$, in analogy to 14 we have*

[TABLE]

We claim that

[TABLE]

where as defined earlier, $\widetilde{\operatorname{\mathbb{E}}}_{x}^{*}=\widetilde{\operatorname{\mathbb{E}}}_{x}^{\psi^{*}}$, and $Y^{*}$ denotes the ground state process. Assuming Eq. 2.62 is true, we first apply the Itô–Krylov formula to Eq. 2.61 to obtain the analogous equation to Eq. 2.60, and then take limits and use Birkhoff’s ergodic theorem to establish Eq. 2.52.

It remains to prove Eq. 2.62. Choose $\epsilon>0$ so that $\beta>\beta-\epsilon>\beta_{c}$, and let $\Psi^{*}_{\beta-\epsilon}$ denote the ground state corresponding to $\Lambda_{\beta-\epsilon}$. We choose a ball $\mathscr{B}$ such that

[TABLE]

Since $f$ vanishes at infinity, and $\Lambda_{\beta}>\Lambda_{\beta-\epsilon}$, there exists a constant $\alpha>1$ and a ball also denoted as $\mathscr{B}$, such that

[TABLE]

Since the ground state processes corresponding to the principal eigenvalues $\Lambda_{\beta}$ and $\Lambda_{\beta-\epsilon}$ are ergodic we have from Lemma 2.7 that

[TABLE]

for all $x\in\mathscr{B}^{c}$ where $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. By Eq. 2.27, the function $\widetilde{\Psi}_{\epsilon}\coloneqq\frac{\Psi^{*}_{\beta-\epsilon}}{\Psi^{*}_{\beta}}$ satisfies

[TABLE]

Applying the Feynman–Kac formula to Eq. 2.66, using Eq. 2.63, it follows as in [10, Lemma 2.1] that $\inf_{\mathbb{R}^{d}}\,\widetilde{\Psi}_{\epsilon}=\min_{\bar{\mathscr{B}}}\,\widetilde{\Psi}_{\epsilon}$. Thus $\widetilde{\Psi}_{\epsilon}$ is bounded away from [math] on ${\mathbb{R}^{d}}$.

Let $\kappa\coloneqq\min_{\mathscr{B}}\frac{\Psi^{*}_{\beta-\epsilon}}{(\Psi^{*}_{\beta})^{\alpha}}$. Then by Eqs. 2.64 and 2.65 we obtain

[TABLE]

Therefore, for some constant $\kappa_{1}$ we have

[TABLE]

Let $\epsilon_{\circ}\coloneqq\frac{1}{2}(\Lambda_{\beta}-\Lambda_{\beta-\epsilon})$. From Eq. 2.63 and exponential Foster–Lyapunov equation Eq. 2.66 we deduce that Eq. 2.54 holds for $\widetilde{\Psi}_{\epsilon}$. Thus the first equation in Eq. 2.62 follows directly from Eqs. 2.54 and 2.67 and the fact that $\inf_{{\mathbb{R}^{d}}}\psi^{*}>-\infty$, while the second one follows by repeating the argument leading to Eq. 2.57. This completes the proof. \qed

Remark 2.1

The assumption that Eq. 2.1 is recurrent in the case that $\Lambda_{\beta}<0$ in Theorem 2.7 (iii) is equivalent to the statement that $\lambda^{\!*}(0)=0$. Note that as shown in [18, Theorem 2.1], unless $\lambda^{\!*}(0)=0$, then Eq. 2.52 does not hold if $\Lambda_{\beta}<0$.

If Eq. 2.1 is not recurrent, then it is possible that $\beta_{c}<0$ and also that $\Lambda_{\beta}<0$ for $\beta\geq 0$. Consider a diffusion with $d=1$, $b(x)=2x$, and $\upsigma(x)=\sqrt{2}$. Then, we have $\mathscr{L}\varphi=-\varphi$ for $\varphi(x)=\frac{1}{2}e^{-x^{2}}$. Thus $\hat{\Lambda}_{0}\leq-1$, where $\hat{\Lambda}_{0}$ denotes the eigenvalue in Eq. 2.5 for $f=0$. Thus $\lambda^{\!*}(0)\leq-1$ by Lemma 2.2 (b). Since the twisted process corresponding to $\varphi$ is exponentially ergodic, we must have $\lambda^{\!*}(0)=-1$ by Theorem 2.1 (c), and thus $\varphi$ is the ground state. Theorem 2.1 (b) then asserts that $\beta\mapsto\Lambda_{\beta}$ is strictly increasing at $\beta=0$. Thus $\beta_{c}<0$. Observe that the ground state diffusion is an Ornstein–Uhlenbeck process having a Gaussian stationary distribution of mean [math] and variance $\nicefrac{{1}}{{2}}$. An easy computation reveals that \mu^{*}\bigl{(}-\langle\nabla\psi^{*},a\nabla\psi^{*}\rangle\bigr{)}=-2 which is smaller than $\lambda^{\!*}(0)$.

The conclusion of Theorem 2.7 (iii) can be sharpened. Consider the controlled diffusion

[TABLE]

Here $v\colon{\mathbb{R}^{d}}\to{\mathbb{R}^{d}}$ is a locally bounded Borel measurable map. Let $\widehat{\mathfrak{U}}_{\mathrm{SM}}$ denote the class of such maps. These are identified with the class of locally bounded stationary Markov controls. Let $\widehat{\mathfrak{U}}_{\mathrm{SSM}}\subset\widehat{\mathfrak{U}}_{\mathrm{SM}}$ be the collection of those $v$ under which the diffusion in Eq. 2.68 is ergodic, and denote by $\widehat{\mu}_{v}$ the associated invariant probability measure. We let $\mathscr{A}_{v}\coloneqq\mathscr{L}+2\langle av,\nabla\rangle$, and use the symbol $\widehat{\operatorname{\mathbb{E}}}^{v}_{x}$ to denote the expectation operator associated with Eq. 2.68.

In order to simplify the notation, we use the norm $\lVert v\rVert_{a}\coloneqq\sqrt{\langle v,av\rangle}$. For $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}$ we define

[TABLE]

and $\overline{\mathscr{J}}_{x}\coloneqq\inf_{v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}}\;\mathscr{J}_{x}(v)$.

Theorem 2.8

Assume that $f\in\mathcal{B}^{+}_{\mathrm{o}}({\mathbb{R}^{d}})$ and $\beta>\beta_{c}$. Then the following hold

If $\Lambda_{\beta}>0$, then we have

[TABLE]

In addition, if $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}$ satisfies $\mathscr{J}_{x}(v)=\overline{\mathscr{J}}_{x}$, then $v=\nabla\psi^{*}_{\beta}$ a.e. 2. 2.

If $\Lambda_{\beta}\leq 0$ and Eq. 2.1 is recurrent then Eq. 2.69 holds, and $v=\nabla\psi^{*}_{\beta}$ is the a.e. unique control in $\widehat{\mathfrak{U}}_{\mathrm{SSM}}$ which satisfies $\mathscr{J}_{x}(v)=\overline{\mathscr{J}}_{x}$. 3. 3.

If $\Lambda_{\beta}<0$ and Eq. 2.1 is not recurrent, then $\overline{\mathscr{J}}_{x}=0$ for all $x\in{\mathbb{R}^{d}}$.

Proof 15

We start with part (a). By Theorem 2.7 (iii), we have $\mathscr{J}_{x}(\nabla\psi^{*}_{\beta})=-\Lambda_{\beta}$ in both of cases (a) and (b). It suffices then to show that if $\mathscr{J}_{x}(v)\leq-\Lambda_{\beta}$ for some $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}$, then $v=\nabla\psi^{*}_{\beta}$ a.e. in ${\mathbb{R}^{d}}$. Let such a control $v$ be given. Then Eq. 2.68 must be positive recurrent under $v$, for otherwise we must have \mathscr{J}_{x}(v)\geq\limsup_{T\to\infty}\;\frac{1}{T}\,\widehat{\operatorname{\mathbb{E}}}^{v}_{x}\bigl{[}\int_{0}^{T}-\beta f(Z_{s})\,\mathrm{d}s\bigr{]}\geq 0. Therefore,

[TABLE]

where $\widehat{\mu}_{v}$, as defined earlier, denotes the invariant probability measure associated with $\mathscr{A}_{v}\coloneqq\mathscr{L}+2\langle v,a\nabla\rangle$. Thus, $\mathscr{J}_{x}(v)$ does not depend on $x$, and dropping this dependence in the notation we let $\mathscr{J}(v)=\mathscr{J}_{x}(v)$. Since $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, it follows by Eq. 2.70 and the definition of $F_{v}$ that there exists a ball $\mathscr{B}$ such that

[TABLE]

By Eq. 2.71, and since $v$ is locally bounded, and $F_{v}$ is integrable with respect to $\widehat{\mu}_{v}$, we can assert the existence of a solution $\breve{\varphi}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ to the Poisson equation

[TABLE]

which is bounded below in ${\mathbb{R}^{d}}$ (see Lemma 3.7.8 (d) in [34]). It follows by Eq. 2.72 that $\Phi\coloneqq\mathrm{e}^{\varphi}$, $\varphi=-\breve{\varphi}$, satisfies

[TABLE]

This shows that \bigl{(}\Phi,-\mathscr{J}(v)\bigr{)} is an eigenpair for $\mathscr{L}^{\breve{F}}$, with $\breve{F}\coloneqq\beta f-\lVert v-\nabla\varphi\rVert^{2}_{a}$. The corresponding twisted process with generator $\tilde{\mathscr{L}}=\mathscr{L}+2\langle a\nabla\varphi,\nabla\rangle$ then satisfies

[TABLE]

Since $\breve{\varphi}$ is bounded below in ${\mathbb{R}^{d}}$ and $\mathscr{J}(v)<0$, Eq. 2.74 shows that the twisted process is positive recurrent. We claim that $-\mathscr{J}(v)$ is the principal eigenvalue of $\mathscr{L}^{\breve{F}}$. Indeed, if $\lambda^{\!*}(\breve{F})<-\mathscr{J}(v)$ then by the proof of Lemma 2.3 and for any $g\in C^{+}_{\mathrm{c}}({\mathbb{R}^{d}})$ we obtain

[TABLE]

for all sufficiently large $n$, where $\uptau_{n}$ denotes the first exit time from $B_{n}$. By first letting $n\to\infty$, and then integrating with respect to $T$ we obtain

[TABLE]

But this contradicts the positive recurrence of the twisted process corresponding to $\tilde{\mathscr{L}}$. Therefore, $-\mathscr{J}(v)$ must be the principal eigenvalue of $\mathscr{L}^{\breve{F}}$, which implies that

[TABLE]

Thus we have shown that $\mathscr{J}_{x}(v)=\mathscr{J}(v)=-\Lambda_{\beta}$. The strict monotonicity of $\lambda^{\!*}$ at $\beta f$ together with Eq. 2.75 imply that $v=\nabla\varphi$ a.e. in ${\mathbb{R}^{d}}$. In turn, Eq. 2.73 and the uniqueness of the ground state imply that $\Phi=\Psi^{*}_{\beta}$, up to a multiplication by a positive constant. Therefore, we have $v=\nabla\psi^{*}_{\beta}$ a.e. in ${\mathbb{R}^{d}}$, and this completes the proof of part (a).

We continue with part (b). The case $\Lambda_{\beta}=0$ is trivial, so assume that $\Lambda_{\beta}<0$. Then Eq. 2.1 is exponentially ergodic by Theorem 2.7(ii). Thus $\Psi^{*}_{\beta}$ is bounded away from [math] in ${\mathbb{R}^{d}}$ by [10, Lemma 2.1]. Let $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}$, and $\breve{\psi}=-\psi^{*}_{\beta}$. We have

[TABLE]

Since $\breve{\psi}$ is bounded above in ${\mathbb{R}^{d}}$, it follows from Eq. 2.76 by a standard argument that

[TABLE]

We next show uniqueness in $\widehat{\mathfrak{U}}_{\mathrm{SSM}}$ of the optimal control $\nabla\psi^{*}_{\beta}$. Let $v\in\widehat{\mathfrak{U}}_{\mathrm{SSM}}$ and suppose $\mathscr{J}_{x}(v)=-\Lambda_{\beta}$. In other words, $\widehat{\mu}_{v}(F_{v})=-\Lambda_{\beta}$. By the Itô–Krylov formula and Fatou’s lemma and since $\breve{\psi}$ is bounded above, we obtain from Eq. 2.76 that

[TABLE]

with

[TABLE]

Dividing Eq. 2.77 by $t$ and taking limits as $t\to\infty$, we obtain $-\widehat{\mu}_{v}(G_{v})+\mathscr{J}_{x}(v)\geq-\Lambda_{\beta}$. Therefore, $\widehat{\mu}_{v}(G_{v})=0$, since $G_{v}$ is nonnegative. Thus $G_{v}=0$ a.e. in ${\mathbb{R}^{d}}$, and since $\widehat{\mu}_{v}$ has a density, this implies that $v=-\nabla\breve{\psi}=\nabla\psi^{*}_{\beta}$ a.e. in ${\mathbb{R}^{d}}$.

We now turn to part (c). It is evident that under the control $v=0$, since the diffusion in Eq. 2.68 is transient and $f$ vanishes at infinity, we have \lim_{t\to\infty}\,\frac{1}{t}\,\widehat{\operatorname{\mathbb{E}}}^{v}_{x}\bigl{[}F_{0}(Z_{t})\bigr{]}=0. It is also clear that under any control $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}\setminus\widehat{\mathfrak{U}}_{\mathrm{SSM}}$ we have \lim_{t\to\infty}\,\frac{1}{t}\,\widehat{\operatorname{\mathbb{E}}}^{v}_{x}\bigl{[}F_{v}(Z_{t})\bigr{]}\geq 0. Suppose that under some $v\in\widehat{\mathfrak{U}}_{\mathrm{SSM}}$, we have

[TABLE]

Then there exists a solution $\breve{\phi}$ to the Poisson equation Eq. 2.72 which is bounded below in ${\mathbb{R}^{d}}$. Thus following the proof of Case 1 in part (a) we obtain by Eq. 2.75 that $\mathscr{J}_{x}(v)\geq-\Lambda_{\beta}$ which is a contradiction. We have therefore shown that $\mathscr{J}_{x}(v)\geq 0$ for all $v\in\widehat{\mathfrak{U}}_{\mathrm{SM}}$, which implies that [math] is the optimal value in the class of controls $\widehat{\mathfrak{U}}_{\mathrm{SM}}$. \qed

Remark 2.2

The assumption that $f$ is nonnegative can be weakened to $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$. From the proof of Theorem 2.2 we note that if $\lambda^{\!*}(f+h)<\lambda^{\!*}(f)$ for some $h\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, then the ground state diffusion corresponding to $\lambda^{\!*}(f)$ is geometrically ergodic. Moreover, due to [5, Proposition 2.3 (vii)] the function $\beta\mapsto\lambda^{\!*}(\beta f)$ is convex for every $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$. Instead of the critical value $\beta_{c}$, we can define a critical value $\lambda_{c}$ by $\lambda_{c}\coloneqq\inf_{\beta\in\mathbb{R}}\,\Lambda_{\beta}$. Then if we replace the condition $\beta>\beta_{c}$ by $\Lambda_{\beta}>\lambda_{c}$ as done in [18], it is evident that $\lambda^{\!*}(\beta f)$ is strictly monotone at $\beta f$ and the results in Theorem 2.7 (iii) and Theorem 2.8 still hold, provided $\Lambda_{\beta}\neq 0$, and the proofs are the same.

The results in Theorem 2.8 (b) can be also stated for nonstationary controls. Consider the controlled diffusion

[TABLE]

Here $U=\{U_{t}\}$ is an ${\mathbb{R}^{d}}-$valued control process which is jointly measurable in $(t,\omega)\in[0,\infty)\times\Omega$, and is nonanticipative: for $t>s$, $W_{t}-W_{s}$ is independent of

[TABLE]

Let $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, not necessarily nonnegative. Assume that $\Lambda_{\beta}>\lambda_{c}$, $\Lambda_{\beta}\leq 0$, and Eq. 2.1 is recurrent (see Remark 2.2). Suppose that under $U$, the diffusion in Eq. 2.78 has a unique weak solution. We claim that

[TABLE]

We can prove this as follows. By 14 we obtain

[TABLE]

for all $u,z\in{\mathbb{R}^{d}}$, and we apply the Itô–Krylov formula and Fatou’s lemma (using the fact that $\breve{\psi}$ is bounded above) with $u=U_{t}$ to obtain analogously to Eq. 2.77 that

[TABLE]

Dividing Eq. 2.79 by $t$ and letting $t\to\infty$, we obtain

[TABLE]

thus proving the claim.

2.4.1 Strong duality

The optimality result in Theorem 2.8 can be strengthened. Consider the class of infinitesimal ergodic occupation measures, i.e., measures $\uppi\in\mathcal{P}({\mathbb{R}^{d}}\times{\mathbb{R}^{d}})$ which satisfy

[TABLE]

with $\mathscr{A}_{u}\coloneqq\mathscr{L}+\langle 2au,\nabla\rangle$. Disintegrate these as $\uppi(\mathrm{d}{x},\mathrm{d}{u})=\eta_{v}(\mathrm{d}{x})\,v(\mathrm{d}{u}\,|\,x)$, and denote this disintegration as $\uppi=\eta_{v}\circledast v$. Let $\hat{v}(x)=\int u\,v(\mathrm{d}{u}\,|\,x)$. Since $\int\lvert u\rvert^{2}\eta(\mathrm{d}{x})\,v(\mathrm{d}{u}\,|\,x)\geq\int\lvert\hat{v}(x)\rvert^{2}\eta(\mathrm{d}{x})$, and $\eta(\mathrm{d}x)\delta_{\hat{v}(x)}(\mathrm{d}u)$ is also an ergodic occupation measure, it is enough to consider the class of infinitesimal ergodic occupation measures $\uppi$ that correspond to a precise control $v$, i.e., a Borel measurable map from ${\mathbb{R}^{d}}$ to ${\mathbb{R}^{d}}$. We denote this class by $\mathscr{M}$. Thus for $\uppi=\eta_{v}\circledast v\in\mathscr{M}$, Eq. 2.80 takes the form $\int_{{\mathbb{R}^{d}}}\mathscr{A}_{v}g(x)\,\eta_{v}(\mathrm{d}{x})=0$. Note that $v$ is not necessarily locally bounded, so this class of controls is, in general, larger than $\widehat{\mathfrak{U}}_{\mathrm{SSM}}$.

In Theorem 2.9 below we use the following simple assertions which are stated as remarks.

Remark 2.3

If $\eta_{v}$ has density $\rho_{v}\in L_{\mathrm{loc}}^{\nicefrac{{d}}{{(d-1)}}}({\mathbb{R}^{d}})$, and $v\in L^{2}({\mathbb{R}^{d}};\eta_{v})$, then

[TABLE]

This can be proved as follows. We mollify $g$ with a smooth mollifier family $\{\chi_{r},\,r>0\}$, so that Eq. 2.80 can applied to the function $g*\chi_{r}$, where ‘$*$’ denotes convolution. Then we separate terms, and applying the Hölder inequality on \bigl{\lvert}\int(\mathscr{L}g-\mathscr{L}(g*\chi_{r}))\rho_{v}\bigr{\rvert}, and using the convergence of $\mathscr{L}(g*\chi_{r})$ to $\mathscr{L}g$ in $L_{\mathrm{loc}}^{d}({\mathbb{R}^{d}})$, we deduce that this term tends to [math] as $r\searrow 0$. Similarly, we apply the Hölder inequality in the form

[TABLE]

Then the first integral on the right hand side is bounded, and the second integral vanishes as $r\searrow 0$ since $g*\chi_{r}$ converges to $g$ uniformly on compact sets.

Remark 2.4

Suppose that the drift $b$ in Eq. 2.1 has at most affine growth. It is then well known that the map $x\mapsto\operatorname{\mathbb{E}}_{x}[\uptau(\mathscr{B}^{c})]$ is inf-compact for any open ball $\mathscr{B}$, provided of course that Eq. 2.1 is positive recurrent. This fact together with the stochastic representation in Eq. 2.18 and Jensen’s inequality, imply that if $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, $\Lambda_{\beta}<0$, and Eq. 2.1 is recurrent, then the ground state $\Psi^{*}_{\beta}$ is inf-compact, and this of course renders Eq. 2.1 positive recurrent. An analogous argument using the ground state diffusion shows that, if $b+a\nabla\Psi^{*}_{\beta}$ has at most affine growth, and $\Lambda_{\beta}>\max\{0,\lambda_{c}\}$ (see Remark 2.2), then $\widetilde{\Psi}=(\Psi^{*}_{\beta})^{-1}$ is inf-compact.

The theorem that follows shows that there is no optimality gap between the primal problem which consists of minimizing $\int F_{u}(x)\uppi(\mathrm{d}x,\mathrm{d}u)$ subject to the constraint Eq. 2.80, and the dual problem which amounts to a maximization over subsolutions of the HJB equation, as described in Section 1. This theorem is stated for $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$ which is not necessarily nonnegative as discussed in Remark 2.2.

Theorem 2.9

Assume that $f\in\mathcal{B}_{\mathrm{o}}({\mathbb{R}^{d}})$, $\Lambda_{\beta}>\lambda_{c}$, and that one of the following conditions holds.

$\Lambda_{\beta}>0$, the coefficients $a$ and $b$ are bounded, and $a$ is uniformly strictly elliptic. 2. 2.

$\Lambda_{\beta}<0$, Eq. 2.1 is recurrent, and $b$ has at most affine growth.

Then any $\uppi=\eta_{v}\circledast v\in\mathscr{M}$, such that $\int_{\mathbb{R}^{d}}F_{v}\,\mathrm{d}\eta_{v}<\infty$, satisfies

[TABLE]

In addition, if $\uppi=\eta_{v}\circledast v\in\mathscr{M}$ is optimal, i.e., if it satisfies $\int_{\mathbb{R}^{d}}F_{v}\,\mathrm{d}\eta_{v}=-\Lambda_{\beta}$, then $v=\nabla\psi^{*}_{\beta}$ a.e. in ${\mathbb{R}^{d}}$ and $\eta_{v}=\mu^{*}_{\beta}$.

Proof 16

We first consider case (i). Since $a$, $b$, and $f$ are bounded, it follows that $\nabla\psi^{*}_{\beta}$ is bounded by [10, Lemma 3.3]. Then $-\psi^{*}_{\beta}$ is inf-compact by Remark 2.4. Recall that $\mathscr{A}_{v}=\mathscr{L}+2\langle a\,v,\nabla\rangle$. We have

[TABLE]

Let $\chi$ be a convex $C^{2}(\mathbb{R})$ function such that $\chi(x)=x$ for $x\geq 0$, $\chi(x)=-1$ for $x\leq-1$, and $\chi^{\prime}$, $\chi^{\prime\prime}$ are positive on $(-1,0)$. Define $\chi_{R}(x)\coloneqq-R+\chi(x+R)$, $R>0$. Then we have from Eq. 2.82 that

[TABLE]

Since $\int\mathscr{A}_{v}g\,\mathrm{d}\eta_{v}=$ for all $g\in C^{\infty}_{\mathrm{c}}({\mathbb{R}^{d}})$, an application of [41, Theorem 2.1] shows that $\eta_{v}$ has a density $\rho_{v}\in L_{\mathrm{loc}}^{\nicefrac{{d}}{{(d-1)}}}({\mathbb{R}^{d}})$. Note that this does not require $a$ or $b$ to be bounded. Therefore, since $\chi_{R}(\psi^{*}_{\beta})+R+1$ has compact support, we have $\int_{\mathbb{R}^{d}}\mathscr{A}_{v}\chi_{R}(\psi^{*}_{\beta})\,\eta_{v}(\mathrm{d}x)=0$ by Remark 2.3. Thus letting $R\to\infty$ in Eq. 2.83, using monotone convergence, we obtain Eq. 2.81.

We next show uniqueness. Let $\uppi=\eta_{v}\circledast v\in\mathscr{M}$ be optimal, and $\uppi_{*}=\eta_{*}\circledast v_{*}$ denote the ergodic occupation measure corresponding to $v_{*}=\nabla\psi^{*}_{\beta}$. Here, $\eta_{*}=\mu^{*}_{\beta}$. Let $\rho_{*}$ denote the density of $\eta_{*}$. Define $\bar{\eta}\coloneqq\frac{1}{2}(\eta_{v}+\eta_{*})$ and $\bar{v}\coloneqq\zeta_{v}v+\zeta_{*}v_{*}$, with $\zeta_{v}$ and $\zeta_{*}$ given by $\zeta_{v}\coloneqq\frac{\rho_{v}}{\rho_{v}+\rho_{*}}$ and $\zeta_{*}\coloneqq\frac{\rho_{*}}{\rho_{v}+\rho_{*}}$, respectively. It is straightforward to verify, using the fact that the drift is affine in the control, that $\bar{\uppi}=\bar{\eta}\circledast\bar{v}$ is in $\mathscr{M}$.

By optimality, we have

[TABLE]

Since $\rho_{*}$ is strictly positive, 16 implies that $\rho_{v}\,\lvert v-v_{*}\rvert=0$ a.e. in ${\mathbb{R}^{d}}$, and thus $v=v_{*}$ on the support of $\eta_{v}$. It is clear that if $v$ is modified outside the support of $\eta_{v}$, then the modified $\eta_{v}\circledast v$ is also an infinitesimal ergodic occupation measure. Therefore $\eta_{v}\circledast v_{*}\in\mathscr{M}$. The uniqueness of the invariant measure of the diffusion with generator $\mathscr{A}_{v_{*}}$ then implies that $\eta_{v}=\eta_{*}$, which in turn implies that $v=\nabla\psi^{*}_{\beta}$ a.e. in ${\mathbb{R}^{d}}$.

We now turn to case (ii). By Remark 2.4, $\psi^{*}_{\beta}$ is inf-compact. Also, as shown in case (i), $\eta_{v}$ has a density $\rho_{v}\in L_{\mathrm{loc}}^{\nicefrac{{d}}{{(d-1)}}}({\mathbb{R}^{d}})$. We write Eq. 2.76 as

[TABLE]

with $\breve{\psi}=-\psi^{*}_{\beta}$. Then we have from Eq. 2.85 that

[TABLE]

Using the inequality $\lVert v+\nabla\breve{\psi}\rVert^{2}_{a}\leq 2\lVert v\rVert^{2}_{a}+2\lVert\nabla\breve{\psi}\rVert^{2}_{a}$, then integrating Eq. 2.86 with respect to $\eta_{v}$, and rearranging terms we obtain

[TABLE]

Thus letting $R\to\infty$ in Eq. 2.87, using monotone convergence, we obtain the energy inequality

[TABLE]

Then Eq. 2.81 follows by letting $R\to\infty$ in Eq. 2.87, using again monotone convergence and Eq. 2.88. Uniqueness follows as in case (i). This completes the proof. \qed

Remark 2.5

The proof of Theorem 2.9 provides a general recipe to prove the lack of an optimality gap in ergodic control problems. Note that the model in [16] is such that $\nabla\psi^{*}_{\beta}$ is bounded, and $a$ is also bounded. Therefore,

[TABLE]

and the proof of Theorem 2.9 goes through even for the more general Hamiltonian $H(x,p)$ in [16].

Remark 2.6

If $\Lambda_{\beta}>\lambda_{c}$, and under some nonanticipative control $U$ the diffusion Eq. 2.78 has a unique weak solution, it was shown in the discussion following Remark 2.2 that $\mathscr{J}_{x}(U)\geq-\Lambda_{\beta}$, provided $\Lambda_{\beta}\leq 0$ and Eq. 2.1 is recurrent. The same conclusion can be drawn if $\Lambda_{\beta}>0$ and under the hypotheses of Theorem 2.9. Define the set of mean empirical measures \bigl{\{}\xi^{U}_{x,t}\,,\;t\geq 0\} of Eq. 2.78 under the control $U$ by

[TABLE]

If $\Lambda_{\beta}>0$, then $F_{u}(x)-\Lambda_{\beta}$ is bounded away from zero for all $x$ outside some compact set, and one can follow the arguments in the proof of [34, Lemma 3.4.6] to show that every limit point in $\mathcal{P}(\overline{{\mathbb{R}^{d}}\times{\mathbb{R}^{d}}})$ (the set of Borel probability measures on the one-point compactification of ${\mathbb{R}^{d}}\times{\mathbb{R}^{d}}$) of a sequence of mean empirical measures $\{\xi^{U_{n}}_{x,{t_{n}}}\,,\;n\in\mathbb{N}\}$ as $t_{n}\to\infty$ takes the form $\delta\uppi+(1-\delta)\uppi_{\infty}$, where $\uppi$ is an infinitesimal ergodic occupation measure and $\uppi_{\infty}(\{\infty\})=1$. Using this property, one can show, by following the argument in the proof of [34, Theorem 3.4.7], that if $\mathscr{J}_{x}(U)\leq-\Lambda_{\beta}$, then the mean empirical measures are necessarily tight in $\mathcal{P}({\mathbb{R}^{d}}\times{\mathbb{R}^{d}})$ and $\delta=1$ in this decomposition. This of course implies that $\mathscr{J}_{x}(U)=-\Lambda_{\beta}$. This argument establishes optimality over the largest possible class of controls $U$.

2.4.2 Differentiability of $\Lambda_{\beta}$

Differentiability of the map $\beta\mapsto\Lambda_{\beta}$ for all $\beta>\beta_{c}$ is established in [18, Proposition 5.4] under the hypothesis that the coefficients $a$, $b$, and $f$ are Lipschitz continuous and bounded in ${\mathbb{R}^{d}}$, but for a more general class of Hamiltonians (see (A1)–(A3) in [18]). These assumptions are used to show that $\nabla\psi^{*}$ is bounded in ${\mathbb{R}^{d}}$, and this is utilized in the proofs.

In the next theorem we demonstrate this differentiability result for the model in this paper which assumes only measurable $b$ and $f$, in which case it is not possible, in general, to obtain gradient estimates and follow the approach in [16, 18, 19]. The first assertion in this theorem should be compared to [18, Proposition 5.4]. Recall the definition $\widetilde{\Psi}_{\epsilon}=\frac{\Psi^{*}_{\beta-\epsilon}}{\Psi^{*}_{\beta}}$ after Eq. 2.65, and let $\widetilde{\psi}_{\epsilon}=\log\widetilde{\Psi}_{\epsilon}$.

Theorem 2.10

Suppose $f\in\mathcal{B}^{+}_{\mathrm{o}}({\mathbb{R}^{d}})$, and that $\beta>\beta_{c}$. Then for all $\epsilon>0$ such that $\beta-\epsilon>\beta_{c}$, we have

[TABLE]

In addition, we have

[TABLE]

Proof 17

Fix some $\epsilon_{1}>0$ such that $\beta-2\varepsilon_{1}>\beta_{c}$, and consider Eq. 2.66. As argued in the proof of Theorem 2.7, the function $\widetilde{\Psi}_{\epsilon}$ is bounded away from [math] on ${\mathbb{R}^{d}}$ for all $\epsilon\in(0,\varepsilon_{1}]$. We recall the notation $\widetilde{\operatorname{\mathbb{E}}}^{\psi^{*}_{\beta}}[\,\cdot\,]=\widetilde{\operatorname{\mathbb{E}}}^{*}[\,\cdot\,]$. Applying the Itô–Krylov formula and Fatou’s lemma to Eq. 2.66 we obtain

[TABLE]

from which the left hand side inequality of Eq. 2.89 follows by an application of Birkhoff’s ergodic theorem. Also the analogous estimate to Eq. 2.54 holds for $\widetilde{\Psi}_{\epsilon}$, which implies that

[TABLE]

The second equality in Eq. 2.89 then follows by first using the technique in the proof of Theorem 2.7 and Eq. 2.91 to establish Eq. 2.56 for $\widetilde{\psi}_{\epsilon}$, $\epsilon\in(0,\varepsilon_{1})$, and then applying the Itô–Krylov formula to the log-transformed equation corresponding to Eq. 2.66 as in Eq. 2.60, and taking limits at $t\to\infty$.

Using the convexity of $\beta\mapsto\Lambda_{\beta}$, we write Eq. 2.89 as

[TABLE]

Fix an open ball $\mathscr{B}\subset{\mathbb{R}^{d}}$, such that

[TABLE]

This is clearly possible since $\epsilon\mapsto\Lambda_{\beta-\epsilon}$ is nonincreasing, $\Lambda_{\beta-2\epsilon_{1}}<\Lambda_{\beta-\epsilon_{1}}$, and $f$ vanishes at infinity. Let $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. Since the ground state process corresponding to $\Lambda_{\beta-\epsilon}$ is exponentially ergodic for $\epsilon<\epsilon_{1}$ by Theorem 2.7, we have

[TABLE]

by Lemma 2.7. Since $\Psi^{*}_{\beta-\epsilon}$ and its inverse are bounded on $\mathscr{B}$, uniformly in $\epsilon\in[-\epsilon_{1},2\epsilon_{1}]$, it follows from Eqs. 2.93 and 2.94 that there exists $\kappa$ such that $\Psi^{*}_{\beta-\epsilon}\leq\kappa\Psi^{*}_{\beta-2\epsilon_{1}}$ for all $\epsilon\in[-\epsilon_{1},\epsilon_{1}]$. Therefore, since the collection \bigl{\{}\Psi^{*}_{\beta-\epsilon}\,,\,\epsilon\in[-\epsilon_{1},\epsilon_{1}]\bigr{\}}, is bounded in $C_{\mathrm{loc}}^{1,\alpha}(\mathscr{B})$, $\alpha>0$, we can use Eq. 2.94 and the dominated convergence theorem to conclude that $\widetilde{\Psi}_{\epsilon}\to 1$ as $\epsilon\searrow 0$. Thus, one more application of the dominated convergence theorem shows that $\mu^{*}_{\beta}(f\,\widetilde{\Psi}_{\epsilon})\to\mu^{*}_{\beta}(f)$ and $\mu^{*}_{\beta}(\widetilde{\Psi}_{\epsilon})\to 1$ as $\epsilon\searrow 0$. This shows that

[TABLE]

We next study the term $\mu^{*}_{\beta+\epsilon}(f)$. Let $\widetilde{\operatorname{\mathbb{E}}}_{x}^{*,\epsilon}$ denote the expectation operator for the ground state diffusion corresponding to $\Lambda_{\beta+\epsilon}$. Since

[TABLE]

it follows by an estimate similar to Eq. 2.93 that $\widetilde{\operatorname{\mathbb{E}}}_{x}^{*,\epsilon}[\mathrm{e}^{\kappa\breve{\uptau}}]\;\leq\;\tfrac{\Psi^{*}_{\beta-2\epsilon_{1}}}{\Psi^{*}_{\beta+\epsilon}}(x)$ for all $x\in\mathscr{B}^{c}$ (see also Theorem 3.1 in Section 3).

We claim that

[TABLE]

Indeed, let $\tilde{\mathscr{B}}$ be a larger ball such that $\bar{\mathscr{B}}\subset\tilde{\mathscr{B}}$. It suffices to exhibit the result for $\tilde{\mathscr{B}}$. For some positive constants $\delta_{i}$, $i=1,2,3$, we have

[TABLE]

and also (see [34, Theorem 2.6.1])

[TABLE]

We use the inequality $\mu^{*}_{\beta+\epsilon}(\tilde{\mathscr{B}})\geq\frac{\delta_{2}}{\delta_{1}+\delta_{3}}$, which follows from the well-known characterization of invariant probability measures due to Has${}^{{}_{{}^{{}^{\prime}}}}\!$minskiĭ [34, Theorem 2.6.9], and which establishes the claim.

It follows from Eq. 2.96 that the corresponding densities $\eta^{*}_{\beta+\epsilon}$ are locally bounded and also bounded away from [math] uniformly in $\epsilon\in[0,\epsilon_{1}]$ by the Harnack inequality (see proof of equation (3.2.6) in [34]). Therefore, standard pde estimates of the Fokker–Planck equation show that this family of densities is locally Hölder equicontinuous [35, Theorem 8.24, p. 202]. Given any $\theta\in(0,1)$ we may enlarge $\mathscr{B}$ so that $\mu^{*}_{\beta}(\mathscr{B})\geq 1-\theta$ and $\lvert f\rvert\leq\theta$ on $\mathscr{B}^{c}$. Let $\bar{\eta}_{\beta}$ be the (uniform) limit of $\eta^{*}_{\beta+\epsilon_{n}}$ on $\mathscr{B}$ along some subsequence $\epsilon_{n}\searrow 0$. Since $\nabla\psi^{*}_{\beta-\epsilon}$ is Hölder equicontinuous on $\mathscr{B}$, uniformly in $\epsilon\in[-\epsilon_{1},\epsilon_{1}]$ as argued earlier, it follows that $\bar{\eta}_{\beta}$ is strictly positive on $\bar{\mathscr{B}}$. It is straightforward to show then that $\bar{\eta}_{\beta}$ is a positive solution of the Fokker–Planck equation for the (adjoint of the) operator $\mathscr{L}+2\langle a\nabla\psi^{*}_{\beta},\nabla\rangle$. By the uniqueness of the invariant probability measure we have $\bar{\eta}_{\beta}=C\eta^{*}_{\beta}$ for some positive constant $C$. Since $\int_{\mathscr{B}}\bar{\eta}_{\beta}(x)\,\mathrm{d}x\leq 1$, we have $C\leq(1-\theta)^{-1}$. Thus, since $\sup_{\epsilon\in[0,\epsilon_{1}]}\,\lVert\eta^{*}_{\beta+\epsilon}\rVert_{\infty}<\infty$, and $\lvert f\rvert<\theta$ on $\mathscr{B}^{c}$, by Fatou’s lemma we obtain

[TABLE]

Since $\theta$ can be selected arbitrarily close to [math], we obtain from Eq. 2.92 that $\lim_{\epsilon\searrow 0}\,\frac{\Lambda_{\beta+\epsilon}-\Lambda_{\beta}}{\epsilon}\leq\mu^{*}_{\beta}(f)$. Combining this with Eqs. 2.94 and 2.95 we obtain Eq. 2.90. \qed

3 Exponential ergodicity and strict monotonicity of principal eigenvalues

In this section we show that exponential ergodicity of Eq. 2.1 is a sufficient condition for the strict monotonicity of the principal eigenvalue. In [17, 9] exponential ergodicity is used to obtain results similar to Theorem 2.1. In these studies the coefficients $a$, $b$, and $f$ are assumed to be $C^{2}$, and this assumption seems hard to waive as the technique used relies on a gradient estimate (see [17, Theorem 3.1] and [9, Lemma 2.4]) which is not available for less regular coefficients. Our approach has allowed us to obtain the results in Section 2 under much weaker hypotheses on the coefficients. Under some additional hypotheses, we show in this section that $\lambda^{\!*}(f)=\mathscr{E}(f)$. Recall the definition of $\lambda^{\prime\prime}(f)$ in Eq. 2.46. It is straightforward to show that $\lambda^{\prime\prime}(f)\geq\mathscr{E}(f)$. We present an example where $\lambda^{\!*}(f)<\mathscr{E}(f)$, and therefore also $\lambda^{\!*}(f)<\lambda^{\prime\prime}(f)$.

Example 3.1

Let $\phi:\mathbb{R}\to\mathbb{R}_{+}$ be a smooth function which is strictly positive on $[-1,1]$ and satisfies $\phi(x)=\mathrm{e}^{-\frac{1}{2}\lvert x\rvert}$ for $\lvert x\rvert\geq 1$. Define

[TABLE]

Then $f(x)=\frac{5}{4}$ for $\lvert x\rvert\geq 1$, and

[TABLE]

Consider the one-dimensional controlled diffusion

[TABLE]

From Eq. 3.1 and Lemma 2.2 (ii) we have $\lambda^{\!*}(f)\leq 1$. It is clear that Eq. 3.2 is a transient process. Therefore, for any initial data $x$

[TABLE]

Hence $\lambda^{\!*}(f)<\mathscr{E}(f)$.

Remark 3.1

Example 3.1* presents a case where the conclusion of [5, Theorem 1.9] fails to hold. Since the operator $\mathscr{L}$ in this example is uniformly elliptic, has bounded coefficients, and $d=1$, the only aspect that makes it different from the class of operators in part (i) of [5, Theorem 1.9], is that it is not self-adjoint.*

Let us start by summarizing some equivalent characterizations of exponential ergodicity.

Theorem 3.1

The following are equivalent.

For some ball $\mathscr{B}_{\circ}$ there exists $\delta_{\circ}>0$ and $x_{\circ}\in\bar{\mathscr{B}}_{\circ}^{c}$ such that $\operatorname{\mathbb{E}}_{x_{\circ}}[\mathrm{e}^{\delta_{\circ}\,\uptau(\mathscr{B}^{c}_{\circ})}]<\infty$. 2. 2.

For every ball $\mathscr{B}$ there exists $\delta>0$ such that $\operatorname{\mathbb{E}}_{x}[\mathrm{e}^{\delta\,\uptau(\mathscr{B}^{c})}]<\infty\;$ for all $x\in\mathscr{B}^{c}$. 3. 3.

For every ball $\mathscr{B}$, there exists a positive function $\mathscr{V}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, with $\inf_{\mathbb{R}^{d}}\,\mathscr{V}>0$, and positive constants $\kappa_{0}$ and $\delta$ such that

[TABLE] 4. 4.

Equation 2.1* is recurrent, and $\lambda^{\!*}(\mathds{1}_{\mathscr{B}^{c}})<1$ for every ball $\mathscr{B}$.*

Proof 18

We first show that (a)$\,\Rightarrow\,$(d). It is clear that (a) implies that Eq. 2.1 is positive recurrent, and that it is enough to prove that $\lambda^{\!*}(\mathds{1}_{\mathscr{B}^{c}})<1$ for any $\mathscr{B}\subset\mathscr{B}_{\circ}$. Let $f=\mathds{1}_{\mathscr{B}^{c}}$, and consider the Dirichlet eigensolutions $(\widehat{\Psi}_{n},\hat{\lambda}_{n})$ in Eq. 2.3. It is easy to see that $\hat{\lambda}_{n}<1$ for all $n$. We claim that $\lambda^{\!*}(f)<1$. If not, then $\hat{\lambda}_{n}\nearrow 1$ as $n\to\infty$, and $\widehat{\Psi}_{n}$ converges to some $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p\geq d$, which satisfies $\mathscr{L}\Psi=\mathds{1}_{\mathscr{B}}\Psi$ on ${\mathbb{R}^{d}}$ and $\Psi(0)=1$. The same argument used in the proof of Lemma 2.2 then shows that \Psi(x)=\operatorname{\mathbb{E}}_{x}\bigl{[}\Psi(X_{\uptau(\mathscr{B}^{c}_{\circ})}\bigr{]}. Therefore, $\Psi$ attains a maximum on $\bar{\mathscr{B}}_{\circ}$, and by the strong maximum principle it must be constant. Thus $\mathscr{L}\Psi=0$ which contradicts the fact that $\Psi(0)=1$.

Next we show that (d)$\,\Rightarrow\,$(c). If $\lambda^{\!*}(f)<1$, for $f=\mathds{1}_{\mathscr{B}^{c}}$, then any limit point $\Psi$ of the Dirichlet eigenfunctions $\widehat{\Psi}_{n}$ as $n\to\infty$ satisfies

[TABLE]

Also by [10, Lemma 2.1 (c)], we have $\inf_{\mathbb{R}^{d}}\,\Psi=\min_{\bar{\mathscr{B}}}\,\Psi>0$. Thus (c) holds with $\delta=1-\lambda^{\!*}(f)$.

That (c)$\,\Rightarrow\,$(b) is well known, and can be shown by a standard application of the Itô–Krylov formula to Eq. 3.3, by which we obtain

[TABLE]

The result then follows by letting $R\to\infty$, and this completes the proof. \qed

We introduce the following hypothesis.

• (H2)

There exists a lower-semicontinuous, inf-compact function $\ell:{\mathbb{R}^{d}}\to[0,\infty)$ such that $\mathscr{E}(\ell)<\infty$, where $\mathscr{E}(\cdot)$ is as defined in Eq. 1.3.

Lemma 3.1

Under (H2), we have $\mathscr{E}(\ell)=\lambda^{\!*}(\ell)$, and there exists a positive $V\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p\geq d$, with $\inf_{{\mathbb{R}^{d}}}V>0$, and $V(0)=1$, satisfying

[TABLE]

In particular, the unique strong solution of Eq. 2.1 is exponentially ergodic.

Proof 19

By Eq. 2.4 we have

[TABLE]

Since $\mathscr{E}(\ell)=\inf_{{\mathbb{R}^{d}}}\mathscr{E}_{x}(\ell)$, Eq. 3.5 implies that

[TABLE]

for some $x\in{\mathbb{R}^{d}}$. The inf-compactness of $\ell$ then implies that the unique strong solution of Eq. 2.1 is positive recurrent. That $\mathscr{E}(\ell)=\lambda^{\!*}(\ell)$, and the existence of a solution $V$ then follow by Theorem 1.4 and Lemma 2.1 in [10], respectively. Exponential ergodicity then follows from Eq. 3.4, using Theorem 3.1. \qed

An application of the Itô–Krylov formula to Eq. 3.4, followed by Fatou’s lemma, shows that

[TABLE]

where $\breve{\uptau}_{r}$, as defined earlier, denotes the first hitting time of the ball $B_{r}$.

The next result shows that (H2) implies (P1).

Theorem 3.2

Assume (H2), and suppose that $f$ is a potential such that $\ell-f$ is inf-compact. Then for any continuous $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ we have

[TABLE]

Proof 20

Let $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$, and $\tilde{f}\coloneqq f-h$. It is easy to see that $\mathscr{E}(f)$ and $\mathscr{E}(\tilde{f})$ are both finite. It is shown in [10, 28] that the Dirichlet eigensolutions $(\widehat{\Psi}_{r},\hat{\lambda}_{r})$ in Eq. 2.3 converge, along some subsequence as $r\to\infty$, to \bigl{(}\Psi^{*},\lambda^{\!*}(f)\bigr{)} which satisfies

[TABLE]

It is also clear that Lemma 2.2 (i) holds for $(\widehat{\Psi}_{n},\hat{\lambda}_{n})$. Now choose a bounded ball $\mathscr{B}$ such that

[TABLE]

This is possible since $\ell-f$ is inf-compact. In view of Eq. 3.6 we note that Eq. 2.12 holds with $f-h-\lambda^{\!*}(f-h)$ replaced by $\ell-\lambda^{\!*}(\ell)$. Thus with the above choice of $\mathscr{B}$, we can justify the passing to the limit in Eq. 2.14, and therefore, we obtain

[TABLE]

with $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. Recall the definition of \bigl{(}\tilde{\Psi}^{*},\lambda^{\!*}(\tilde{f})\bigr{)} in Eq. 2.31. A similar argument also gives

[TABLE]

In fact, the above relations hold for any bounded domain $D\supset\mathscr{B}$ with $\breve{\uptau}=\uptau(D^{c})$.

Suppose that $\lambda^{\!*}(f)=\lambda^{\!*}(\tilde{f})$. Then

[TABLE]

Thus if we multiply $\Psi^{*}$ with a suitable positive constant such that $\Psi^{*}-\tilde{\Psi}^{*}$ is nonnegative in $\mathscr{B}$ and attains a minimum of [math] in $\mathscr{B}$, it follows from Eqs. 3.8 and 3.9 that $\Psi^{*}-\tilde{\Psi}^{*}$ is nonnegative in ${\mathbb{R}^{d}}$. Since Eq. 2.34 holds, and we conclude exactly as in the proof of Theorem 2.2 that $\lambda^{\!*}(\tilde{f})<\lambda^{\!*}(f)$.

Next we show that $\lambda^{\!*}(f)=\mathscr{E}_{x}(f)$ for all $x\in{\mathbb{R}^{d}}$. We have already established the strict monotonicity of $\lambda^{\!*}(f)$ at $f$, and therefore, Theorem 2.1 applies. Hence for any continuous $g$ with compact support we have from [40, Theorem 1.3.10] that

[TABLE]

where $\mu^{*}$ denotes the invariant measure of the twisted process $Y^{*}$ satisfying Eq. 2.25. Let $\tilde{\mathscr{B}}$ be a ball such that $f(x)-\lambda^{\!*}(f)<\ell(x)-\lambda^{\!*}(\ell)$ for $x\in\tilde{\mathscr{B}}^{c}$. Thus from Eq. 3.4 we obtain

[TABLE]

with \kappa=\max_{\tilde{\mathscr{B}}}\bigl{(}\lvert f\rvert+\lvert\ell\rvert+\lvert\lambda^{\!*}\rvert+\lambda^{\!*}(\ell)\bigr{)}\,V. Applying the Itô–Krylov formula to Eq. 3.11 followed by Fatou’s lemma we obtain

[TABLE]

for some constant $\kappa^{\prime}$, where in the last inequality we have used Eq. 3.10. Taking logarithms on both sides of the preceding inequality, then dividing by $T$, and letting $T\to\infty$, we obtain $\lambda^{\!*}(f)\geq\mathscr{E}_{x}(f)$ for all $x\in{\mathbb{R}^{d}}$. Combining this with Eq. 3.7 results in equality. \qed

Remark 3.2

Continuity of $h$ is superfluous in Theorem 3.2. The result holds if $h$ is a non-trivial, nonnegative measurable function, vanishing at infinity.

Corollary 3.1

Under the assumptions of Theorem 3.2, for any potential $\tilde{f}\lneqq f$, we have $\lambda^{\!*}(\tilde{f})<\lambda^{\!*}(f)$.

Proof 21

Note that for any cut-off function $\chi$ we have $\lambda^{\!*}(\tilde{f})\leq\lambda^{\!*}(\chi\tilde{f}+(1-\chi)f)$. Then the result follows from Theorems 3.2 and 3.2. \qed

Remark 3.3

In Theorem 3.2 we can replace the assumption that $f$ is bounded from below in ${\mathbb{R}^{d}}$ by the hypothesis that $\ell-\lvert f\rvert$ is inf-compact.

Let us now discuss the exponential ergodicity and show that this implies (H2).

Proposition 3.1

Let $\ell:{\mathbb{R}^{d}}\to\mathbb{R}$ be inf-compact, and suppose $\phi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ is bounded below in ${\mathbb{R}^{d}}$ and satisfies

[TABLE]

then $\mathscr{E}_{x}(\ell)<\infty\;$ for all $x\in{\mathbb{R}^{d}}$.

Proof 22

Let $\Phi(x)=\exp(\phi(x))$. Then $\inf_{{\mathbb{R}^{d}}}\Phi>0$, and Eq. 3.12 gives

[TABLE]

Now apply the Itô–Krylov formula to Eq. 3.13 followed by Fatou’s lemma to obtain

[TABLE]

Taking logarithm on both sides, diving by $T$ and letting $T\to\infty$, we obtain $\mathscr{E}_{x}(\ell)<\infty$. \qed

Example 3.2

Let $a=\frac{1}{2}I$ and $b(x)=b_{1}(x)+B(x)$ where $B$ is bounded and

[TABLE]

Then we take $\phi(x)=\theta\,|x|^{\alpha}$ for $\lvert x\rvert\geq 1\,,\theta\in(0,1)$. It is easy to check that for a suitable choice of $\theta\in(0,1)$, Eq. 3.12 holds for $\ell(x)\sim\lvert x\rvert^{2\alpha-2}$.

Remark 3.4

Equation 3.12* is a stronger condition than strict monotonicity of $\lambda^{\!*}(f)$ at $f$. In fact, Eq. 3.12 might not hold in many important situations. For instance, if $a$ and $b$ are both bounded, and $a$ is uniformly elliptic, then it is not possible to find inf-compact $\ell$ satisfying Eq. 3.12. Otherwise, we can find a finite principal eigenvalue for the operator $\mathscr{L}^{\ell}$, by a same method as in Eq. 3.7, which would contradict [5, Proposition 2.6].*

Even though Eq. 3.12 does not hold for bounded $a$ and $b$, strict monotonicity of $\lambda^{\!*}(f)$ at $f$ can be asserted under suitable hypotheses. This is the subject of the following theorem.

Theorem 3.3

Let $\mathscr{V}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ such that $\inf_{{\mathbb{R}^{d}}}\mathscr{V}>0$, satisfying

[TABLE]

for some compact set $\mathscr{K}$ and positive constants $\kappa_{0}$ and $\gamma$. Let $f$ be a nonnegative bounded measurable function with $\limsup_{x\to\infty}\,f(x)<\gamma$. Then for any $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$, we have $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)=\mathscr{E}_{x}(f)$ for all $x\in{\mathbb{R}^{d}}$.

Proof 23

Let $\tilde{f}=f-h$. Suppose $\lambda^{\!*}(\tilde{f})=\lambda^{\!*}(f)$. Applying an argument similar to Eq. 3.7 we can find $\Psi^{*}$ and $\tilde{\Psi}^{*}$ that satisfy

[TABLE]

Let $\mathscr{K}_{0}\supset\mathscr{K}$ be any compact set such that $f<\gamma$ on $\mathscr{K}_{0}^{c}$. If $\breve{\uptau}$ denotes the first hitting time to the compact set $\mathscr{K}_{0}$, then by an application of the Itô–Krylov formula to Eq. 3.14 we obtain

[TABLE]

We next use the fact that if $\mathscr{L}$ corresponds to a recurrent diffusion and $f$ is nonnegative then $\lambda^{\!*}(f)\geq 0$. Indeed, in this case we have $\mathscr{L}\Psi^{*}\leq\lambda^{\!*}(f)\Psi^{*}$. If $\lambda^{\!*}(f)\leq 0$, this implies that $\Psi^{*}(X_{t})$ is a nonnegative supermartingale and since it is integrable, it converges a.s. Since the process is recurrent, this implies that $\Psi^{*}$ must equal to a constant, which, in turn, necessitates that $\lambda^{\!*}(f)=0$ (and $f=0$). Thus, since $\lambda^{\!*}(f)\geq 0$, an argument similar to the proof of Lemma 2.2 (ii) shows that

[TABLE]

for $x\in\mathscr{K}_{0}^{c}$. Therefore, applying the strong maximum principle as in Theorem 3.2, we obtain $h\,\tilde{\Psi}^{*}=0$ which is a contradiction since $h\neq 0$ and $\tilde{\Psi}^{*}>0$. Thus we have $\lambda^{\!*}(f-h)<\lambda^{\!*}(f)$. That $\lambda^{\!*}(f)=\mathscr{E}_{x}(f)$ for all $x\in{\mathbb{R}^{d}}$ follows by an argument similar to the one used in the proof Theorem 3.2. \qed

Example 3.3

Suppose $a=\frac{1}{2}I$, where $I$ denotes the identity matrix, and

[TABLE]

With $\mathscr{V}(x)=\exp(\lvert x\rvert)$ for $\lvert x\rvert\geq 1$, we have

[TABLE]

4 Risk-sensitive control

In this section we apply the results developed in the previous sections to the risk-sensitive control problem. As mentioned earlier, we establish the existence and uniqueness of solutions to the risk-sensitive HJB equation, and use this to completely characterize the optimal Markov controls (see Theorems 4.1 and 4.2). Another interesting result is the continuity of the controlled principal eigenvalue with respect to the stationary Markov controls. This is done in Theorem 4.3. We first introduce the control problem.

4.1 The controlled diffusion model

Consider a controlled diffusion process $X=\{X_{t},\,t\geq 0\}$ which takes values in the $d$-dimensional Euclidean space $\mathbb{R}^{d}$, and is governed by the Itô equation

[TABLE]

All random processes in Eq. 4.1 live in a complete probability space $(\Omega,\mathfrak{F},\operatorname{\mathbb{P}})$. The process $W$ is a $d$-dimensional standard Wiener process independent of the initial condition $X_{0}$. The control process $U$ takes values in a compact, metrizable set $\mathbb{U}$, and $U_{t}(\omega)$ is jointly measurable in $(t,\omega)\in[0,\infty)\times\Omega$. The set $\mathfrak{U}$ of admissible controls consists of the control processes $U$ that are non-anticipative: for $s<t$, $W_{t}-W_{s}$ is independent of

[TABLE]

We impose the following standard assumptions on the drift $b$ and the diffusion matrix $\upsigma$ to guarantee existence and uniqueness of solutions.

• (B1)

Local Lipschitz continuity: The functions $b\colon\mathbb{R}^{d}\times\mathbb{U}\to\mathbb{R}^{d}$ and $\upsigma\colon\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ are continuous, and satisfy

[TABLE]

for some constant $C_{R}>0$ depending on $R>0$.

• (B2)

Affine growth condition: For some $C_{0}>0$, we have

[TABLE]

• (B3)

Nondegeneracy: Assumption (A3) in Subsection 1.1 holds.

It is well known that under (B1)–(B3), for any admissible control there exists a unique solution of Eq. 4.1 [34, Theorem 2.2.4]. We define the family of operators $\mathcal{L}_{u}\colon C^{2}(\mathbb{R}^{d})\mapsto C(\mathbb{R}^{d})$, where $u\in\mathbb{U}$ plays the role of a parameter, by

[TABLE]

The risk-sensitive criterion

Let $\mathfrak{C}$ denote the class of functions $c(x,u)$ in $C({\mathbb{R}^{d}}\times\mathbb{U},\mathbb{R}_{+})$ that are locally Lipschitz in $x$ uniformly with respect to $u\in\mathbb{U}$. We let $c\in\mathfrak{C}$ denote the running cost function, and for any admissible control $U\in\mathfrak{U}$, we define the risk-sensitive objective function $\mathscr{E}^{U}_{x}(c)$ by

[TABLE]

We also define $\Lambda^{\!*}_{x}\coloneqq\inf_{U\in\mathfrak{U}}\,\mathscr{E}^{U}_{x}(c)$.

4.2 Relaxed controls

We adopt the well-known relaxed control framework [34]. According to this relaxation, a stationary Markov control is a measurable map from ${\mathbb{R}^{d}}$ to $\mathcal{P}(\mathbb{U})$, the latter denoting the set of probability measures on $\mathbb{U}$ under the Prokhorov topology. Let $\mathfrak{U}_{\mathrm{SM}}$ denote the class of all such stationary Markov controls. A control $v\in\mathfrak{U}_{\mathrm{SM}}$ may be viewed as a kernel on $\mathcal{P}(\mathbb{U})\times{\mathbb{R}^{d}}$, which we write as $v(\mathrm{d}{u}\!\mid\!x)$. We say that a control $v\in\mathfrak{U}_{\mathrm{SM}}$ is precise if it is a measurable map from ${\mathbb{R}^{d}}$ to $\mathbb{U}$. We extend the definition of $b$ and $c$ as follows. For $v\in\mathfrak{U}_{\mathrm{SM}}$ we let

[TABLE]

It is easy to see from (B2) and Jensen’s inequality that

[TABLE]

For $v\in\mathfrak{U}_{\mathrm{SM}}$, consider the relaxed diffusion

[TABLE]

It is well known that under $v\in\mathfrak{U}_{\mathrm{SM}}$ Eq. 4.3 has a unique strong solution [42], which is also a strong Markov process. It also follows from the work in [41] that under $v\in\mathfrak{U}_{\mathrm{SM}}$, the transition probabilities of $X$ have densities which are locally Hölder continuous. Thus $\mathscr{L}_{v}$ defined by

[TABLE]

for $f\in C^{2}(\mathbb{R}^{d})$, is the generator of a strongly-continuous semigroup on $C_{b}(\mathbb{R}^{d})$, which is strong Feller. We let $\operatorname{\mathbb{P}}_{x}^{v}$ denote the probability measure and $\operatorname{\mathbb{E}}_{x}^{v}$ the expectation operator on the canonical space of the process under the control $v\in\mathfrak{U}_{\mathrm{SM}}$, conditioned on the process $X$ starting from $x\in\mathbb{R}^{d}$ at $t=0$. We denote by $\mathfrak{U}_{\mathrm{SSM}}$ the subset of $\mathfrak{U}_{\mathrm{SM}}$ that consists of stable controls, i.e., under which the controlled process is positive recurrent, and by $\mu_{v}$ the invariant probability measure of the process under the control $v\in\mathfrak{U}_{\mathrm{SSM}}$.

Definition 4.1

For $v\in\mathfrak{U}_{\mathrm{SM}}$ and a locally bounded measurable function $f\colon{\mathbb{R}^{d}}\to\mathbb{R}$, we let $\lambda^{\!*}_{v}(f)$ denote the principal eigenvalue of the operator $\mathscr{L}_{v}^{f}\coloneqq\mathscr{L}_{v}+f$ on ${\mathbb{R}^{d}}$ (see Definition 2.1).

We also adapt the notation in Eq. 2.4 to the control setting, and define

[TABLE]

We refer to $\mathscr{E}^{v}(f)$ as the risk-sensitive average of $f$ under the control $v$.

Recall the risk-sensitive objective function $\mathscr{E}_{x}^{U}$ defined in Eq. 4.2 and the optimal value $\Lambda^{\!*}$. We say that a stationary Markov control $v\in\mathfrak{U}_{\mathrm{SM}}$ is optimal (for the risk-sensitive criterion) if $\mathscr{E}^{v}_{x}(c_{v})=\Lambda^{\!*}_{x}$ for all $x\in{\mathbb{R}^{d}}$, and we let $\mathfrak{U}_{\mathrm{SM}}^{*}$ denote the class of these controls.

4.3 Optimal Markov controls and the risk-sensitive HJB

We start with the following assumption.

Assumption 4.1** (uniform exponential ergodicity)**

There exists an inf-compact function $\ell\in C({\mathbb{R}^{d}})$ and a positive function $\mathscr{V}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, satisfying $\inf_{\mathbb{R}^{d}}\,\mathscr{V}>0$, such that

[TABLE]

for some constant $\bar{\kappa}$, and a compact set $\mathscr{K}$.

It is easy to see that for $\bar{\kappa}_{\circ}\coloneqq\frac{\bar{\kappa}}{\min_{{\mathbb{R}^{d}}}\,\mathscr{V}}$ we obtain from Eq. 4.4 that

[TABLE]

and therefore, applying the Itô–Krylov formula, we have $\mathscr{E}^{v}_{x}(\ell)\leq\bar{\kappa}_{\circ}$ for any stationary Markov control $v\in\mathfrak{U}_{\mathrm{SM}}$, and all $x\in{\mathbb{R}^{d}}$.

Example 4.1

Let $\upsigma$ be bounded and $b:{\mathbb{R}^{d}}\times\mathbb{U}\to{\mathbb{R}^{d}}$ be such that

[TABLE]

Then as seen in Example 3.2, $\mathscr{V}(x)=\exp(\theta\,\lvert x\rvert^{\alpha})$, for $\lvert x\rvert\geq 1$, satisfies Eq. 4.4 for sufficiently small $\theta>0$, and $\ell(x)\sim\lvert x\rvert^{2\alpha-2}$. Note that $\alpha=2$ and $\upsigma=I$ is considered in [23].

We introduce the class of running costs $\mathcal{C}_{\ell}$ defined by

[TABLE]

The first important result of this section is the following.

Theorem 4.1

Suppose Assumption 4.1 holds, and $c\in\mathcal{C}_{\ell}$. Then $\Lambda^{\!*}=\Lambda^{\!*}_{x}$ does not depend on $x$, and there exists a positive solution $V\in C^{2}({\mathbb{R}^{d}})$ satisfying

[TABLE]

In addition, if $\;\overline{\mathfrak{U}}_{\mathrm{SM}}\subset\mathfrak{U}_{\mathrm{SM}}$ denotes the class of Markov controls $v$ which satisfy

[TABLE]

then the following hold.

$\overline{\mathfrak{U}}_{\mathrm{SM}}\subset\mathfrak{U}_{\mathrm{SM}}^{*}$, and it holds that $\lambda^{\!*}_{v}(c_{v})=\Lambda^{\!*}$ for all $v\in\overline{\mathfrak{U}}_{\mathrm{SM}}$; 2. 2.

$\mathfrak{U}_{\mathrm{SM}}^{*}\subset\overline{\mathfrak{U}}_{\mathrm{SM}}\,$; 3. 3.

Equation 4.5* has a unique positive solution in $C^{2}({\mathbb{R}^{d}})$ (up to a multiplicative constant).*

Proof 24

Using a standard argument (see [26, 28, 10]) we can find a pair $(V,\hat{\lambda})\in C^{2}({\mathbb{R}^{d}})\times\mathbb{R}$, with $V>0$ on ${\mathbb{R}^{d}}$, and $V(0)=1$, that satisfies

[TABLE]

This is obtained as a limit of Dirichlet eigensolutions (\widehat{V}_{n},\hat{\lambda}_{n})\in\bigl{(}\mathscr{W}_{\mathrm{loc}}^{2,p}(B_{n})\cap C(\bar{B}_{n})\bigr{)}\times\mathbb{R}, for any $p>d$, satisfying $\widehat{V}_{n}>0$ on $B_{n}$, $\widehat{V}_{n}=0$ on $\partial B_{n}$, $\widehat{V}_{n}(0)=1$, and

[TABLE]

For $v\in\overline{\mathfrak{U}}_{\mathrm{SM}}$ we have

[TABLE]

By Corollary 2.1 we obtain $\hat{\lambda}\geq\lambda^{\!*}_{v}(c_{v})$. Also by Theorem 3.2 we have $\lambda^{\!*}_{v}(c_{v})=\mathscr{E}^{v}_{x}(c_{v})$ for all $x\in{\mathbb{R}^{d}}$. Combining these estimates with Eq. 4.6 we obtain

[TABLE]

This of course shows that $\hat{\lambda}=\lambda^{\!*}_{v}(c_{v})=\Lambda^{\!*}_{x}$ for all $x\in{\mathbb{R}^{d}}$, and also proves part (a).

We continue with part (b). By Theorem 3.2 we have

[TABLE]

In turn, by Lemma 2.4 there exists a unique eigenfunction $\Psi_{v}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$ which is associated with the principal eigenvalue $\lambda^{\!*}_{v}(c_{v})$ of the operator $\mathscr{L}_{v}^{c_{v}}=\mathscr{L}_{v}+c_{v}$. Since $\hat{\lambda}=\lambda^{\!*}_{v}(c_{v})$ for all $v\in\overline{\mathfrak{U}}_{\mathrm{SM}}$ by part (a), it follows by Eq. 4.7 that

[TABLE]

By Eq. 4.8 and Lemma 2.2 (ii), and since Eq. 4.3 is recurrent, we have

[TABLE]

and all sufficiently large balls $\mathscr{B}$ centered at [math], where $\breve{\uptau}=\uptau(\mathscr{B}^{c})$, as usual.

Since the Dirichlet eigenvalues satisfy $\hat{\lambda}_{n}<\hat{\lambda}=\Lambda^{\!*}$ for all $n\in\mathbb{N}$, the Dirichlet problem

[TABLE]

with $\alpha_{n}>0$, has a unique solution $\varphi_{n}\in\mathscr{W}_{\mathrm{loc}}^{2,p}(B_{n})\cap C(\bar{B}_{n})$, for any $p\geq 1$ [4, Theorem 1.9] (see also [43, Theorem 1.1 (ii)]). We choose $\alpha_{n}$ as follows: first select $\tilde{\alpha}_{n}>0$ such that the solution $\varphi_{n}$ of Eq. 4.11 with $\alpha_{n}=\tilde{\alpha}_{n}$ satisfies $\varphi_{n}(0)=1$, and then set $\alpha_{n}=\min(1,\tilde{\alpha}_{n})$. Passing to the limit in Eq. 4.11 as $n\to\infty$ along a subsequence, we obtain a nonnegative solution $\Phi\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$ of

[TABLE]

It is evident from the construction that if $\alpha=0$ then $\Phi(0)=1$. On the other hand, if $\alpha>0$, then necessarily $\Phi$ is positive on ${\mathbb{R}^{d}}$. Let $\hat{v}\in\mathfrak{U}_{\mathrm{SM}}$ be a selector from the minimizer of Eq. 4.12. If $\alpha>0$, then Eq. 4.12 implies that there exists $h\in C_{\mathrm{o}}^{+}({\mathbb{R}^{d}})$ such that $\lambda^{\!*}_{\hat{v}}(c_{\hat{v}}+h)\leq\Lambda^{\!*}$. Since $\lambda^{\!*}_{\hat{v}}(c_{\hat{v}})=\mathscr{E}^{\hat{v}}_{x}(c_{\hat{v}})$ for all $x\in{\mathbb{R}^{d}}$ by Theorem 3.2, and $\mathscr{E}^{\hat{v}}_{x}(c_{\hat{v}})\geq\Lambda^{\!*}$, then, in view of Corollary 2.1, this contradicts Eq. 4.8 and the convexity of $\lambda^{\!*}_{\hat{v}}$. Therefore, we must have $\alpha=0$. Let $\bar{v}\in\mathfrak{U}_{\mathrm{SM}}^{*}$. Applying the Itô–Krylov formula to Eq. 4.11 we obtain

[TABLE]

and for all $x\in B_{n}\setminus\mathscr{B}$, where $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. Using the argument in the proof of [10, Lemma 2.11], we obtain

[TABLE]

Comparing Eq. 4.10 and Eq. 4.13, it follows that, given any $\bar{v}\in\mathfrak{U}_{\mathrm{SM}}^{*}$, we can scale $\Psi_{\bar{v}}$ by a positive constant so that it touches $\Phi$ from above at some point in $\bar{\mathscr{B}}$. However, $\bar{v}$ satisfies

[TABLE]

by Eq. 4.12. Thus we have

[TABLE]

and it follows by the strong maximum principle that $\Phi=\Psi_{\bar{v}}$ for all $\bar{v}\in\mathfrak{U}_{\mathrm{SM}}^{*}$. Since $\overline{\mathfrak{U}}_{\mathrm{SM}}\subset\mathfrak{U}_{\mathrm{SM}}^{*}$ by part (a), it then follows by Eq. 4.9 that $V=\Psi_{\bar{v}}$ for all $\bar{v}\in\mathfrak{U}_{\mathrm{SM}}^{*}$. Thus we have

[TABLE]

This proves the verification of optimality result in part (b).

Suppose now that $\tilde{V}\in C^{2}({\mathbb{R}^{d}})$ is a positive solution of

[TABLE]

Let $\tilde{v}\in\mathfrak{U}_{\mathrm{SM}}$ be a selector from the minimizer of Eq. 4.14. We have $\lambda^{\!*}_{\tilde{v}}(c_{\tilde{v}})=\mathscr{E}^{\tilde{v}}_{x}(c_{\tilde{v}})\geq\Lambda^{\!*}$ for all $x\in{\mathbb{R}^{d}}$ by Theorem 3.2 and the definition of $\Lambda^{\!*}$, and $\lambda^{\!*}_{\tilde{v}}(c_{\tilde{v}})\leq\Lambda^{\!*}$ by Corollary 2.1. Thus $\mathscr{E}^{\tilde{v}}_{x}(c_{\tilde{v}})=\Lambda^{\!*}$ for all $x\in{\mathbb{R}^{d}}$, which implies that $\tilde{v}\in\mathfrak{U}_{\mathrm{SM}}^{*}$. Then $\tilde{V}=\Psi_{\tilde{v}}$ by the uniqueness of the latter. Therefore, $\tilde{V}=\Psi_{\tilde{v}}=V$ by part (b). This completes the proof. \qed

As mentioned in Remark 3.4 the existence of an inf-compact $\ell$ in Assumption 4.1 is not possible when $a$ and $b$ are bounded. So we consider the following alternative assumption.

Assumption 4.2

There exists a function $\mathscr{V}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, such that $\inf_{{\mathbb{R}^{d}}}\mathscr{V}>0$, a compact set $\mathscr{K}$, and positive constants $\kappa_{0}$ and $\gamma$, satisfying

[TABLE]

A similar assumption is used in [26] where the author has obtained only the existence of the solution $V$ to the HJB, and an optimal control. Also it is shown in [26] that there exists a constant $\gamma_{1}$, depending on $\gamma$, such that if $\lVert c\rVert_{\infty}<\gamma_{1}$, then Eq. 4.15 below has a solution. We improve these results substantially by proving uniqueness of the solution $V$, and verification of optimality.

Theorem 4.2

Under Assumption 4.2, there exists a positive solution $V\in C^{2}({\mathbb{R}^{d}})$ satisfying

[TABLE]

Let $\;\overline{\mathfrak{U}}_{\mathrm{SM}}\subset\mathfrak{U}_{\mathrm{SM}}$ be as in Theorem 4.1. Then (a) and (b) of Theorem 4.1 hold, and Eq. 4.15 has a unique positive solution in $C^{2}({\mathbb{R}^{d}})$ up to a multiplicative constant.

Proof 25

Part (a) follows exactly as in the proof of Theorem 4.1.

By Theorems 3.3 and 2.7 for any $v\in\mathfrak{U}_{\mathrm{SM}}$ there exists a unique eigenpair $(\Psi_{v},\lambda^{*}_{v})$ for $\mathscr{L}_{v}^{c_{v}}$. In addition,

[TABLE]

The rest follows as in Theorem 4.1. \qed

4.4 Continuity results

It is known from [34] that the set of relaxed stationary Markov controls $\mathfrak{U}_{\mathrm{SM}}$ is compactly metrizable (see also [44] for a detailed construction of this topology). In particular $v_{n}\to v$ in $\mathfrak{U}_{\mathrm{SM}}$ if and only if

[TABLE]

for all $f\in L^{1}({\mathbb{R}^{d}})\cap L^{2}({\mathbb{R}^{d}})$ and $g\in C_{b}({\mathbb{R}^{d}}\times\mathbb{U})$. For $v\in\mathfrak{U}_{\mathrm{SM}}$ we denote by $(\Psi_{v},\lambda^{\!*}_{v}(f))$ the principal eigenpair of the operator $\mathscr{L}_{v}^{f}$, i.e.,

[TABLE]

When $f=c_{v}$, we occasionally drop the dependence on $c_{v}$ and denote the eigenvalue as $\lambda^{\!*}_{v}=\lambda^{\!*}_{v}(c_{v})$. The next result concerns the continuity of $\lambda^{\!*}_{v}$ with respect to stationary Markov controls, and extends the result in [5, Proposition 9.2]. The continuity result in [5, Proposition 9.2] is established with respect to the $L^{\infty}$ norm convergence of the coefficients, whereas Theorem 4.3 that follows asserts continuity under a much weaker topology.

Theorem 4.3

Assume one of the following.

Assumption 4.1* holds, and $c\in\mathcal{C}_{\beta\ell}$ for some $\beta\in(0,1)$.* 2. 2.

Assumption 4.2* holds.*

Then the map $v\mapsto\lambda^{\!*}_{v}$ is continuous.

Proof 26

We demonstrate the result under (i). For case (ii) the proof is analogous. Let $v_{n}\to v$ in the topology of Markov controls. Let $(\Psi_{n},\lambda^{\!*}_{n})$ be the principal eigenpair which satisfies

[TABLE]

where the equality $\lambda^{\!*}_{n}=\mathscr{E}^{v_{n}}(c_{v_{n}})$ is a consequence of Theorems 3.2 and 3.1. It is obvious that $\lambda^{\!*}_{n}\geq 0$ for all $n$.

Since $\ell(\cdot)-\max_{u\in\mathbb{U}}c(\cdot,u)$ is inf-compact, we can find a constant $\kappa_{1}$ such that $\max_{u\in\mathbb{U}}c(x,u)\leq\kappa_{1}+\ell(x)$. Recall that $\mathscr{E}^{v}(\ell)<\bar{\kappa}_{\circ}$ for all $v\in\mathfrak{U}_{\mathrm{SM}}$ (as shown in the paragraph after Assumption 4.1), and this implies that $\lambda^{\!*}_{n}\leq\kappa_{1}+\bar{\kappa}_{\circ}$ for all $n$. Thus $\{\lambda^{\!*}_{n}\,\colon n\geq 1\}$ is bounded. Therefore, passing to a subsequence we may assume that $\lambda^{\!*}_{n}\to\lambda^{\!*}$ as $n\to\infty$. To complete the proof we only need to show that $\lambda^{\!*}=\lambda^{\!*}_{v}$. Since $\Psi_{n}(0)=1$ for all $n$, and the coefficients $b_{v_{n}}$, and $c_{v_{n}}$ are uniformly locally bounded, applying Harnack’s inequality and Sobolev’s estimate we can find $\Psi\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$, $p\geq 1$, such that $\Psi_{n}\to\Psi$ weakly in $\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{d}})$. Therefore, by [34, Lemma 2.4.3] and Eq. 4.16, we obtain

[TABLE]

By Corollary 2.1 we have $\lambda^{\!*}\geq\lambda^{\!*}_{v}$.

Let $\mathscr{B}\supset\mathscr{K}$ be an open ball such that $\lvert c(x,u)-\lambda^{*}\rvert\leq\beta\ell(x)$ for all $(x,u)\in\mathscr{B}^{c}\times\mathbb{U}$, and $R>0$ be large enough so that $\mathscr{B}\subset B_{R}$. Let $\breve{\uptau}=\uptau(\mathscr{B}^{c})$. Applying the Itô–Krylov formula to Eq. 4.17, we obtain

[TABLE]

for any $T>0$. Since

[TABLE]

and $\Psi$ in bounded in $\mathscr{B}^{c}\cap B_{R}$, for every fixed $R$, letting $T\to\infty$ in Eq. 4.18 we have

[TABLE]

26* which also holds, possibly for a larger ball $\mathscr{B}$, if we replace $v$ and $\lambda^{\!*}$ with $v_{n}$ and $\lambda^{\!*}_{n}$, respectively, shows that, for some constant $\tilde{\kappa}$, we have \Psi_{n}(x)\leq\tilde{\kappa}\bigl{(}\mathscr{V}(x)\bigr{)}^{\beta} for all $n\in\mathbb{N}$, and $x\in\mathscr{B}^{c}$. Therefore, \Psi(x)\leq\tilde{\kappa}\bigl{(}\mathscr{V}(x)\bigr{)}^{\beta} for all $x\in\mathscr{B}^{c}$.*

We write

[TABLE]

The left hand side of Eq. 4.21 and the first term on the right hand side both converge to \operatorname{\mathbb{E}}_{x}^{v}\bigl{[}\mathrm{e}^{\int_{0}^{\breve{\uptau}}\ell(X_{s})\,\mathrm{d}{s}}\bigr{]} as $R\to\infty$, by monotone convergence. Thus we have

[TABLE]

On the other hand Assumption 4.1 implies that

[TABLE]

We proceed as in the proof of Theorem 2.7. Let $\Gamma(R,m)\coloneqq\{x\in\partial B_{R}\colon\Psi(x)\geq m\}$ for $m\geq 1$. Since $\Psi\leq\tilde{\kappa}\mathscr{V}^{\beta}$ on $\mathscr{B}^{c}$, we have \mathscr{V}^{\beta-1}\leq\bigl{(}\tfrac{\Psi}{\tilde{\kappa}}\bigr{)}^{1-\frac{1}{\beta}} on $\mathscr{B}^{c}$, and, therefore,

[TABLE]

Thus, using Eq. 4.23, we obtain

[TABLE]

and by first letting $R\to\infty$, using Eq. 4.22, and then $m\to\infty$, it follows that the left hand side of 26 vanishes as $R\to\infty$. Therefore, letting $R\to\infty$ in Eq. 4.20, we obtain

[TABLE]

It then follows by Corollary 2.3 that $\lambda^{\!*}=\lambda^{\!*}_{v}$, and this completes the proof. \qed

Remark 4.1

Following the proof of Theorem 4.3 we can obtain the following continuity result which should be compared with [5, Proposition 9.2 (ii)]. Consider a sequence of operators $\mathscr{L}_{n}^{f_{n}}$ with coefficients $(a_{n},b_{n},f_{n})$, where $b_{n}$, $f_{n}$ are locally bounded uniformly in $n$, and $\inf_{n}(\inf_{{\mathbb{R}^{d}}}f_{n})>-\infty$. The coefficients $a_{n}$ and $b_{n}$ are assumed to satisfy (A1)–(A3) uniformly in $n$. Assume that $a_{n}\to a$ in $C_{\mathrm{loc}}({\mathbb{R}^{d}})$, and $b_{n}\to b$ and $f_{n}\to f$ weakly in $L^{1}_{\mathrm{loc}}({\mathbb{R}^{d}})$. Moreover we suppose that one of the following hold.

There exists an inf-compact function $\ell\in C({\mathbb{R}^{d}})$ and $\mathscr{V}\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, with $\inf_{{\mathbb{R}^{d}}}\mathscr{V}>0$, such that $\mathscr{L}_{n}\mathscr{V}\;\leq\;\bar{\kappa}\,\mathds{1}_{\mathscr{K}}-\ell\mathscr{V}$ a.e. on ${\mathbb{R}^{d}}$ for some constant $\bar{\kappa}$, and a compact set $\mathscr{K}$. In addition, $\beta\ell-\sup_{n}f_{n}$ is inf-compact for some $\beta\in(0,1)$. 2. 2.

The sequence $\mathscr{L}_{n}$ satisfies Eq. 3.14 for all $n$, $\lim_{n\to\infty}\lVert f^{-}_{n}\rVert_{\infty}=\lVert f^{-}\rVert_{\infty}$, and

[TABLE]

Then the principal eigenvalue $\lambda^{\!*}(f_{n})$ converges to $\lambda^{\!*}(f)$ as $n\to\infty$.

As an application of Theorem 4.3 we have the following existence result for the risk-sensitive control problem under (Markovian) risk-sensitive type constraints.

Theorem 4.4

Assume one of the following.

Assumption 4.1* holds, and $c,r_{1},\dotsc,r_{m}\in\mathcal{C}_{\beta\ell}$ for some $\beta\in(0,1)$.* 2. 2.

Assumption 4.2* holds, and $r_{1},\dotsc,r_{m}\in\mathfrak{C}$ satisfy*

[TABLE]

In addition, suppose that $K_{i}$, $i=1,\dotsc,m$, are closed subsets of $\mathbb{R}$, and that there exists $\hat{v}\in\mathfrak{U}_{\mathrm{SM}}$ such that $\mathscr{E}^{\hat{v}}(r_{i,\hat{v}})\in K_{i}$ for all $i$, where we use the usual notation r_{i,v}(x)\coloneqq r_{i}\bigl{(}x,v(x)\bigr{)}.

Then the following constrained minimization problem admits an optimal control in $\mathfrak{U}_{\mathrm{SM}}$

[TABLE]

Proof 27

Let $v_{n}\in\mathfrak{U}_{\mathrm{SM}}$ be a sequence of controls along which the constraints are met, and $\mathscr{E}^{v_{n}}(c_{v_{n}})$ converges to its infimum. Since $\mathfrak{U}_{\mathrm{SM}}$ is compact under the topology of Markov controls, we may assume, without loss of generality, that $v_{n}$ converges to some $\bar{v}\in\mathfrak{U}_{\mathrm{SM}}$ as $n\to\infty$. By Theorem 4.3 we know that $v\mapsto\lambda^{\!*}_{v}(c_{v})$, and $v\mapsto\lambda^{\!*}_{v}(r_{i,v})$, $i=1,\dotsc,m$, are continuous maps, and that $\mathscr{E}^{v}(c_{v})=\lambda^{\!*}_{v}(c_{v})$, and $\mathscr{E}^{v}(r_{i,v})=\lambda^{\!*}_{v}(r_{i,v})$ for $i=1,\dotsc,m$. It follows that the constraints are met at $\bar{v}$. Therefore, $\bar{v}$ is an optimal Markov control for the constrained problem. \qed

Another application of Theorem 4.3 is a following characterization of $\lambda^{\!*}$ which provides a positive answer to [5, Conjecture 1.8] for a certain class of $a,b$ and $f$. In Theorem 4.5 below, we consider the uncontrolled generator $\mathscr{L}$ in Section 3. Let us introduce the following definition from [5]

[TABLE]

Recall the definition of $\lambda^{\prime\prime}$ in Eq. 2.46. From [5, Theorem 1.7], under (A1)–(A2), we have $\lambda^{\!*}(f)\leq\lambda^{\prime}(f)\leq\lambda^{\prime\prime}(f)$ whenever $f$ is bounded above. It is conjectured in [5, Conjecture 1.8] that for bounded $a$, $b$, and $f$, one has $\lambda^{\prime}(f)=\lambda^{\prime\prime}(f)$. It should be noted from Example 3.1 that $\lambda^{\!*}(f)$ could be strictly smaller than $\lambda^{\prime\prime}(f)$. The following result complements those in [5, Theorems 1.7 and 1.9].

Theorem 4.5

For a potential $f$ the following are true.

Suppose that $\mathscr{E}_{x}(f)<\infty$. Then under (A1)–(A3) we have

[TABLE] 2. 2.

Let $\mathscr{L}$, $\mathscr{V}$ and $\gamma$ satisfy Eq. 3.14, and suppose that $\sup_{{\mathbb{R}^{d}}}(f+\lVert f^{-}\rVert_{\infty})<\gamma$. Then $\lambda^{\!*}(f)\;=\;\lambda^{\prime\prime}(f)$. 3. 3.

Let $\mathscr{L}$, $\mathscr{V}$ and $\ell$ satisfy Eq. 4.4, and suppose that $\beta\ell-f$ is inf-compact for some $\beta\in(0,1)$. Then $\lambda^{\!*}(f)=\lambda^{\prime\prime}(f)$.

Proof 28

We first show (i). By [5, Theorem 1.7 (ii)] we have $\lambda^{\!*}(f)\leq\lambda^{\prime}(f)$. Let $\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})\cap L^{\infty}({\mathbb{R}^{d}})$, $\varphi>0$, be such that

[TABLE]

Recall that $\uptau_{n}$ is the exit time from the open ball $B_{n}(0)$. Therefore, applying the Itô–Krylov formula, we obtain

[TABLE]

Since $\mathscr{E}_{x}(f)$ is finite, letting $n\to\infty$ in Eq. 4.25, taking logarithms on both sides, dividing by $T$ and then letting $T\to\infty$ we obtain $\lambda\leq\mathscr{E}_{x}(f)$. This implies $\lambda^{\prime}(f)\leq\mathscr{E}_{x}(f)$. Now suppose $\varphi\in\mathscr{W}_{\mathrm{loc}}^{2,d}({\mathbb{R}^{d}})$, with $\inf_{\mathbb{R}^{d}}\,\varphi>0$, satisfies

[TABLE]

Repeating the analogous calculation as above, we obtain $\lambda\geq\mathscr{E}_{x}(f)$, which implies that $\mathscr{E}_{x}(f)\leq\lambda^{\prime\prime}(f)$.

Next we prove (ii). Since $\lambda^{\!*}(f+c)=\lambda^{\!*}(f)+c$ for any constant $c$, we may replace $f$ by $f+\lVert f^{-}\rVert_{\infty}$. Therefore, $f$ is non-negative and $\lVert f\rVert_{\infty}<\gamma$. By (i) we have $\lambda^{\!*}(f)\leq\lambda^{\prime\prime}(f)$. Let $\chi_{n}\colon{\mathbb{R}^{d}}\to[0,1]$ be a cut-off function such that $\chi_{n}(x)=1$ for $\lvert x\rvert\leq n$, and $\chi_{n}(x)=0$ for $\lvert x\rvert\geq n+1$. Define $f_{n}\coloneqq\chi_{n}\,f+(1-\chi_{n})\lVert f\rVert_{\infty}$. Let \bigl{(}\Psi^{*}_{n},\lambda^{\!*}(f_{n})\bigr{)} denote the principal eigenpair of $\mathscr{L}^{f_{n}}$. By Remark 4.1 we have $\lambda^{\!*}(f_{n})\to\lambda^{\!*}(f)$ as $n\to\infty$. Thus to complete the proof it is enough to show that $\inf_{{\mathbb{R}^{d}}}\Psi_{n}>0$, which implies that $\lambda^{\!*}(f_{n})=\lambda^{\prime\prime}(f_{n})\geq\lambda^{\prime\prime}(f)$ for all $n$, and thus $\lambda^{\!*}(f)\geq\lambda^{\prime\prime}(f)$. Note that $\lambda^{\!*}(f_{n})\leq\mathscr{E}(f_{n})\leq\lVert f\rVert_{\infty}$ for all $n$. Now fix $n$ and let $\breve{\uptau}_{n}$ be the first hitting time to the ball $B_{n}$. Then applying the Itô–Krylov formula to

[TABLE]

together with Fatou’s lemma, we have

[TABLE]

for all $x\in\overline{B}^{c}_{n+1}(0)$. Hence $\inf_{{\mathbb{R}^{d}}}\Psi^{*}_{n}>0$ which completes the proof.

The proof of (iii) is completely analogous to the proof of part (ii). Since $\beta\ell-f^{+}$ is inf-compact, we can find $g\colon{\mathbb{R}^{d}}\to\mathbb{R}_{+}$, such that $\lim_{\lvert x\rvert\to\infty}g(x)=\infty$, and $\beta\ell-f^{+}-g$ is inf-compact. We let $f_{n}=\chi_{n}f+(1-\chi_{n})(g+f^{+})$. Note that

[TABLE]

is inf-compact. On the other hand, $f_{n}\geq f$ for all $n$. The rest follows as part (ii). \qed

Acknowledgements

The research of Ari Arapostathis was supported in part by the Army Research Office through grant W911NF-17-1-001, in part by the National Science Foundation through grant DMS-1715210, and in part by the Office of Naval Research through grant N00014-16-1-2956. The research of Anup Biswas was supported in part by an INSPIRE faculty fellowship, and a DST-SERB grant EMR/2016/004810.