Practical criteria for R-positive recurrence of unbounded semigroups

TL;DR

This paper leverages recent advances in quasi-stationary distributions to establish practical criteria ensuring geometric convergence of normalized unbounded semigroups, simplifying the analysis of their long-term behavior.

Contribution

It introduces new, easily applicable criteria for R-positive recurrence of unbounded semigroups based on quasi-stationary distribution theory.

Findings

Derived general criteria for geometric convergence

Simplified analysis of unbounded semigroup behavior

Enhanced understanding of R-positive recurrence

Abstract

The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow to deduce effortlessly general criteria for the geometric convergence of normalized unbounded semigroups.

Equations

$$\mu P_{n}f={\int }_E{P}_{n}f(x)\mu (\mathrm{d}x).$$

$$P_{n_1}({\psi }_1\text{mathbbm}{1}_A)(x)\ge c_1\nu (A){\psi }_1(x).$$

$$\underset{x\in K}{\inf }{\psi }_2(x)/{\psi }_1(x)>0,\underset{x\in E}{\sup }{\psi }_2(x)/{\psi }_1(x)\le 1,$$

$$P_1{\psi }_1(x)\le {\theta }_1{\psi }_1(x)+c_2\text{mathbbm}{1}_K(x){\psi }_1(x),\forall x\in E,$$

$$P_1{\psi }_2(x)\ge {\theta }_2{\psi }_2(x),\forall x\in E.$$

$$\underset{n\in {\mathbb{Z}}_{+}}{\sup }\frac{{\sup }_{y\in K}{P}_n{\psi }_1(y)/{\psi }_1(y)}{{\inf }_{y\in K}{P}_n{\psi }_1(y)/{\psi }_1(y)}\le c_3.$$

$$P_n(\text{mathbbm}{1}_K{\psi }_1)>0.$$

$$\left|\frac{\mu P_{n}f}{\mu P_n{\psi }_1}-{\nu }_P(f)\right|\le C{\alpha }^n\frac{\mu ({\psi }_1)}{\mu ({\psi }_2)},\forall n\in {\mathbb{Z}}_{+}.$$

$$\left|{\theta }_0^{-n}\mu P_{n}f-\mu (\eta ){\nu }_P(f)\right|\le C^{\prime }{\beta }^n\mu ({\psi }_1).$$

$$P_n{\psi }_1(x)/{\psi }_1(x)\le c{P}_n{\psi }_2(x)/{\psi }_1(x)\le c{P}_n{\psi }_2(x)/{\psi }_2(x)$$

$$P_n{\psi }_2(x)/{\psi }_2(x)\le P_n{\psi }_1(x)/{\psi }_2(x)\le \underset{K}{\sup }\frac{{\psi }_1}{{\psi }_2}{P}_n{\psi }_1(x)/{\psi }_1(x).$$

$$Q_n(f)=\frac{P_n(f{\psi }_1)}{({\theta }_1+c_2{)}^n{\psi }_1},\forall n\ge 0,\Vert f{\Vert }_{\infty }\le 1.$$

$$\left|\frac{\upsilon Q_{n}g}{\upsilon Q_n\text{mathbbm}1}-{\nu }_{QSD}(g)\right|\le C{\alpha }^n\frac{\Vert g{\Vert }_{\infty }}{\upsilon ({\phi }_2)}.$$

$$\left|{\theta }_Q^{-n}\frac{Q_n(g{\eta }_Q)(x)}{{\eta }_Q(x)}-{\nu }_{QSD}(g{\eta }_Q)\right|\le C^{\prime }{\tilde{\alpha }}^n\frac{1}{{\eta }_Q(x)}.$$

$$\left|{\theta }_0^{-n}{P}_{n}f(x)-\eta (x){\nu }_P(f)\right|\le {\zeta }_n\psi (x)$$

$$R_{n}g(x)=\frac{{\theta }_0^{-n}}{\eta (x)}{P}_n(\eta g)(x),\forall x\in E^{\prime }\text{ and }n\ge 0.$$

$$\left|R_{n}g(x)-{\nu }_R(g)\right|\le {\zeta }_n\frac{\psi (x)}{\eta (x)}.$$

$$R_1{V}_0(x)\le \rho V_0(x)+C_R\text{mathbbm}{1}_K(x),\forall x\in E^{\prime },$$

$$V_0=\sum _{k=0}^{m-1}{\lambda }^{-k}{R}_k\left(\frac{\psi }{\eta }\right)$$

$$R_{n_1^{\prime }}\text{mathbbm}{1}_A(x)\ge c_R{\nu }_R(A),\forall x\in K.$$

$$P_{n_1}(\psi \text{mathbbm}{1}_A)(x)\ge c_R{\theta }_0^{n_1}\eta (x){\nu }_R{R}_{n_1^{\prime \prime }}\left(\text{mathbbm}{1}_K\text{mathbbm}{1}_A\frac{\psi }{\eta }\right),\forall x\in K.$$

$$P_1{\psi }_1(x)\le {\theta }_0\rho {\psi }_1(x)+C_R\text{mathbbm}{1}_K(x)\eta (x)\le {\theta }_0\rho {\psi }_1(x)+\frac{C_R}{{\sup }_E|\eta |/{\psi }_1}\text{mathbbm}{1}_K(x){\psi }_1(x),\forall x\in E^{\prime }.$$

$${\psi }_1(x)=\sum _{k=0}^{m-1}(\lambda {\theta }_0{)}^{-k}{P}_k\psi (x),$$

$$P_1{\psi }_1(x)=\lambda {\theta }_0{\psi }_1(x)-\lambda {\theta }_0\psi (x)+(\lambda {\theta }_0{)}^{1-m}{P}_m\psi \le \lambda {\theta }_0{\psi }_1(x),\forall x\in E\setminus E^{\prime }.$$

$$\left|\frac{\mu P_{t}f}{\mu P_t{\psi }_1}-{\nu }_P(f)\right|\le C^{\prime \prime }{e}^{-\gamma t}\frac{\mu ({\psi }_1)}{\mu ({\psi }_2)},\forall t\in [0,+\infty ),$$

$$\left|e^{-{\lambda }_{0}t}\mu P_{t}f-\mu (\eta ){\nu }_P(f)\right|\le C^{\prime \prime \prime }{e}^{-{\gamma }^{\prime }t}\mu ({\psi }_1),\forall t\in [0,+\infty ).$$

$$\left|\frac{\mu P_{t+n}f}{\mu P_{t+n}{\psi }_1}-{\nu }_P(f)\right|\le C{\alpha }^n\frac{\mu P_t{\psi }_1}{\mu P_t{\psi }_2}\le \frac{C\bar{c}}{\alpha \underline{c}}{\alpha }^{n+t}\frac{\mu ({\psi }_1)}{\mu ({\psi }_2)},$$

$$\left|{\theta }_0^{-n}{P}_{t+n}f(x)-P_t\eta (x){\nu }_P(f)\right|\le C^{\prime }{\beta }^n{P}_t{\psi }_1(x)\le \frac{C^{\prime }\bar{c}}{\beta }{\beta }^{n+t}{\psi }_1(x).$$

$$\left|e^{-{\lambda }_{0}t}{P}_{t}f(x)-\eta (x){\nu }_P(f)\right|\le \frac{C^{\prime }\bar{c}}{\beta }{\beta }^t{\psi }_1(x)$$

$$P_{n}f(x)={\mathbb{E}}_x\left(\prod _{k=1}^{n}G(X_k)\text{mathbbm}{1}_{X_k\in E}f(X_n)\right),$$

$$P_1{\psi }_1(x)\le C\mathbb{E}\left(e^{(1+a)|F(x)+{\xi }_1|}\right)\le C^{\prime }{\psi }_1(x)\exp \left(-a\left(|x|-p|F(x)|\right)\right),$$

Full text

Keywords: R-positivity; quasi-stationary distributions; mixing properties; Foster-Lyapunov criteria

1 Introduction

Let $E$ be a measurable space and $(P_{n},n\in\mathbb{Z}_{+})$ be a positive semigroup of operators on the space $L^{\infty}(\psi_{1})$ to itself, where $\psi_{1}:E\rightarrow(0,+\infty)$ is measurable and $L^{\infty}(\psi_{1})$ is the set of measurable $f:E\rightarrow\mathbb{R}$ such that $|f|/\psi_{1}$ is bounded, endowed with the norm $\|f\|_{\psi_{1}}=\||f|/\psi_{1}\|_{\infty}$. We define the dual action of $(P_{n},n\in\mathbb{Z}_{+})$ on non-negative measures $\mu$ on $E$ such that $\mu(\psi_{1})<+\infty$ as

[TABLE]

Our aim is to provide sufficient conditions for the existence of $\theta_{0}>0$ such that $(\theta_{0}^{-n}P_{n})_{n\in\mathbb{N}}$ converges geometrically toward a non-trivial limit.

In this setting, given $c$ such that $P_{1}\psi_{1}\leq c\psi_{1}$, the operators $Q_{n}=\frac{P_{n}(\cdot\psi_{1})}{c^{n}\psi_{1}}$ defines a sub-Markov semigroup corresponding to a stochastic process with killing. The asymptotic behavior of such semigroups is the subject of the theory of quasi-stationary distributions based on various tools, including the theory of $R$-recurrent Markov chains [31, 29, 28, 17], spectral theoretic results (e.g. Krein-Rutman theorem [13], spectral theory of symetric operators [8, 24], or other general criteria of convergence of normalized semigroups such as the work of Birkhoff [7] and its extensions) and Doeblin’s conditions and Foster-Lyapunov criteria [9, 10]. In this note, we apply the results of [10] to the semigroup $(Q_{n},n\in\mathbb{Z}_{+})$ to give a necessary and sufficient condition for the existence of a nonnegative eigenfunction $\eta$ of $P_{1}$ with eigenvalue $\theta_{0}$ and the geometric convergence of $\theta_{0}^{-n}P_{n}$. We also extend these results to continuous-time semigroups. In particular, our results provide practical criteria for the general theory of $R$-positive recurrence of unbounded semigroups as developed in [29, Section 6.2] and [28]. The notion of $R$-positive recurrence has strong implications for the study of penalized Markov processes [14, 15], of the long time behaviour of Markov branching processes (see for instance [20, 21, 22, 6, 23, 11, 5, 3, 4]), of non-conservative PDEs (see e.g. [1, 2] and references therein) and of measure-valued Pólya processes and reinforced processes [25].

The recent article [2] proposes similar criteria for $R$-positive recurrence of continous-time semigroups with nice applications to growth-fragmentation equations. The extent of our results and approaches sensibly differ. Concerning the results, our criteria apply to a larger class of semigroups including non-irreducible ones (see Remark 2 below). Concerning the approaches, the authors of [2] make use of tools developed in the proofs of [10] adapted to the semigroup setting. We show here how these $R$-positivity criteria can be directly derived as corollaries of the results of [10], applied to the sub-Markov semigroup $(Q_{n},n\in\mathbb{Z}_{+})$. This approach also has the advantage to allow one to deduce with little extra effort sufficient criteria for the convergence of unbounded semigroups from the abundant theory of sub-Markov processes (cf. e.g. [13, 12, 32, 18, 24, 19]). Note that a similar approach has been used in [5] to describe the asymptotic behaviour of the growth-fragmentation equation using Krein-Rutman theorem and other criteria for $R$-positivity. Finally, the authors of [2] also establish a counterpart assuming the existence of a positive eigenfunction of the semigroup and using the approach of [9]. In Theorem 2.2, we extend this counterpart by allowing the eigenfunction to vanish and exhibit the link with the classical theory of $V$-ergodicity [27, 16].

Section 2 is devoted to the statement and the proof of our main results. In Section 3, we provide two applications of these general results to penalized semigroups associated to perturbed (discrete-time) dynamical systems (Subsection 3.1) and diffusion processes (Subsection 3.2).

2 Main result

We first introduce the assumptions on which our results are based. We state them following the same structure as Assumption (E) in [10] to emphasize their similarity.

Condition (G). There exist positive real constants $\theta_{1},\theta_{2},c_{1},c_{2},c_{3}$, an integer $n_{1}\geq 1$, two functions $\psi_{1}:E\rightarrow(0,+\infty)$, $\psi_{2}:E\rightarrow\mathbb{R}_{+}$ and a probability measure $\nu$ on a measurable subset $K$ of $E$ such that

• (G1)

(Local Dobrushin coefficient). $\forall x\in K$ and all measurable $A\subset K$,

[TABLE]

• (G2)

(Global Lyapunov criterion). We have $\theta_{1}<\theta_{2}$ and

[TABLE]

• (G3)

(Local Harnack inequality). We have

[TABLE]

• (G4)

(Aperiodicity). For all $x\in K$, there exists $n_{4}(x)$ such that for all $n\geq n_{4}(x)$,

[TABLE]

Theorem 2.1**.**

Assume that Condition (G) holds true. Then there exist a positive measure $\nu_{P}$ on $E$ such that $\nu_{P}(\psi_{1})=1$ and $\nu_{P}(\psi_{2})>0$, and two constants $C<+\infty$ and $\alpha\in(0,1)$ such that, for all measurable functions $f:E\rightarrow\mathbb{R}$ satisfying $|f|\leq\psi_{1}$ and all positive measures $\mu$ on $E$ such that $\mu(\psi_{1})<+\infty$ and $\mu(\psi_{2})>0$,

[TABLE]

In addition, there exist $\theta_{0}>0$ such that $\nu_{P}P_{n}=\theta_{0}^{n}\nu_{P}$ and a function $\eta:E\rightarrow\mathbb{R}_{+}$ such that $\theta_{0}^{-n}P_{n}\psi_{1}$ converges uniformly and geometrically toward $\eta$ in $L^{\infty}(\psi_{1})$ and such that $P_{1}\eta=\theta_{0}\eta$ and $\nu_{P}(\eta)=\nu_{P}(\psi_{1})=1$. Moreover, there exist two constants $C^{\prime}>0$ and $\beta\in(0,1)$ such that, for all measurable functions $f:E\rightarrow\mathbb{R}$ satisfying $|f|\leq\psi_{1}$ and all positive measures $\mu$ on $E$ such that $\mu(\psi_{1})<+\infty$,

[TABLE]

Remark 1*.*

Note that (G2) implies that $P_{n}\psi_{1}\leq cP_{n}\psi_{2}$ on $K$ for all $n\geq 0$ and some constant $c>0$ (see [10, Lemma 9.6]). Hence we have, for all $x\in K$,

[TABLE]

and

[TABLE]

Therefore, replacing $\psi_{1}$ by $\psi_{2}$ in (G1) and/or (G3) give equivalent versions of Condition (G).

Proof.

Assumption (G2) implies that $P_{1}\psi_{1}\leq(\theta_{1}+c_{2})\psi_{1}$, so that $Q_{1}f:=\frac{P_{1}(f\psi_{1})}{(\theta_{1}+c_{2})\psi_{1}}$ defines a submarkovian kernel generating the semigroup $(Q_{n})_{n\in\mathbb{N}}$ defined by

[TABLE]

It is straightforward to check that this semigroup satisfies conditions (E1-E4) in [10] with $\varphi_{1}=1$ and $\varphi_{2}=\psi_{2}/\psi_{1}$, using $\theta_{1}/(\theta_{1}+c_{2})$ in place of $\theta_{1}$, $\theta_{2}/(\theta_{1}+c_{2})$ in place of $\theta_{2}$ and $c_{1}/(\theta_{1}+c_{2})^{n_{1}}$ in place of $c_{1}$. Using Theorem 2.1 in this reference applied to $Q_{n}$, we deduce that there exist constants $C>0,\alpha\in(0,1)$ and a probability measure $\nu_{QSD}$ on $E$ such that, for all bounded measurable functions $g:E\rightarrow\mathbb{R}$ and all probability measures $\upsilon$ such that $\upsilon(\varphi_{2})>0$,

[TABLE]

Setting $\nu_{P}(dx)=\frac{1}{\psi_{1}(x)}\nu_{QSD}(dx)$, $\mu(dx)=\frac{1}{\psi_{1}(x)}\upsilon(dx)$ and $f=g\,\psi_{1}$, one obtains (2.1). Similarly, Theorem 2.5 of [10] for $Q_{n}$ states that there exist $\theta_{Q}>0$ such that $\nu_{QSD}Q_{n}=\theta_{Q}^{n}\nu_{QSD}$ and a function $\eta_{Q}:E\rightarrow\mathbb{R}_{+}$ such that $\theta_{Q}^{-n}Q_{n}\mathbbm{1}$ converges uniformly and geometrically toward $\eta_{Q}$ in $L^{\infty}$ and such that $Q_{1}\eta_{Q}=\theta_{Q}\eta_{Q}$. Setting $\eta=\eta_{Q}\psi_{1}$ and $\theta_{0}=\theta_{Q}(\theta_{1}+c_{2})$ gives the result on geometric convergence of $\theta_{0}^{-n}P_{n}\psi_{1}$ to $\eta$ in $L^{\infty}(\psi_{1})$.

It remains to prove (2.2). Note that it is sufficient to prove it for any $\mu=\delta_{x}$. If $\eta(x)=0$, this is implied by the above geometric convergence. If $\eta(x)>0$, then $\eta_{Q}(x)>0$ and the convergence of [10, Theorem 2.7] applied to $Q_{n}$ implies that there exists $C^{\prime}<+\infty$ and $\tilde{\alpha}\in(0,1)$ such that, for all measurable $g:E\rightarrow\mathbb{R}$ satisfying $|g|\leq 1/\eta_{Q}$,

[TABLE]

Multiplying both sides by $\eta_{Q}(x)\psi_{1}(x)$ and setting $f=g\eta_{Q}\psi_{1}$ ends the proof of (2.2). ∎

Whether Assumption (G) is necessary for (2.1) is still an open problem up to our knowldge. However, if one assumes that there exists a positive eigenfunction $\eta$ such that (2.2) holds true, then one recovers easily Assumption (G) by applying the classical counterpart of Forster-Lyapunov criteria for conservative semigroups. Here, the conservative semigroup is the one associated to the $h$-tranform of $P_{n}$ defined by $R_{n}f:=\frac{\theta_{0}^{-n}}{\eta}P_{n}(\eta f)$ (which is called $Q$-process in the sub-Markovian case, cf. e.g. [26]). The only difficulty in the proof of the following theorem is that $\eta$ may vanish on some subset of $E$.

Theorem 2.2**.**

Assume that there exist a positive function $\psi:E\rightarrow(0,+\infty)$ and a non-negative eigenfunction $\eta\in L^{\infty}(\psi)$ of $P_{1}$ for the eigenvalue $\theta_{0}>0$, such that

[TABLE]

is satisfied for all $x\in E$ and all measurable functions $f:E\rightarrow\mathbb{R}$ such that $|f|\leq\psi$, where $(\zeta_{n})_{n\geq 0}$ is some positive sequence converging to [math]. Then Assumption (G) is satisfied with $\psi_{2}=\eta$ and with some function $\psi_{1}\in L^{\infty}(\psi)$ such that $\psi\in L^{\infty}(\psi_{1})$.

Proof.

We define $E^{\prime}=\{x\in E,\ \eta(x)>0\}$ and introduce the conservative semigroup $R$ on functions $g:E^{\prime}\rightarrow\mathbb{R}$ such that $|g(x)|\leq\psi(x)/\eta(x)$ defined by

[TABLE]

Applying (2.3) to $f=g\eta$ and setting $\nu_{R}(dx)=\eta(x)\nu_{P}(dx)$, we deduce that, for all $x\in E^{\prime}$ and all measurable function $g:E^{\prime}\rightarrow\mathbb{R}$ such that $|g|\leq\psi/\eta$

[TABLE]

This is the classical $V$-uniform ergodicity condition (with $V=\psi/\eta$), for which necessary and sufficient conditions are well-known. First, it implies $V$-uniform geometric ergodicity, i.e. one can replace $\zeta_{n}$ by $C\,\beta^{n}$ for some $C>0,\beta\in(0,1)$ in the above equation (see for instance Proposition 15.2.3 in [16]). Second, we deduce using for example Theorem 15.2.4(b) in [16] that, for any integer $m$ such that $C^{1/m}\beta<1$ and any $\lambda,\rho$ such that $C^{1/m}\beta\leq\lambda<\rho<1$, there exist $d,C_{R}<+\infty$ such that

[TABLE]

with

[TABLE]

and $K:=\{\psi/\eta\leq d\}\cap E^{\prime}$ is an accessible small set for $R$. This last property means that there exists a probability measure $\nu_{R}$ on $E^{\prime}$ and a constant $c_{R}>0$ such that, for all $A\subset K$ measurable,

[TABLE]

for some constant integer $n^{\prime}_{1}\geq 1$. Since $K$ is accessible, there exists $n^{\prime\prime}_{1}\geq 0$ such that $a:=\nu_{R}R_{n^{\prime\prime}_{1}}\mathbbm{1}_{K}>0$. Setting $n_{1}=n^{\prime}_{1}+n^{\prime\prime}_{1}$, it then follows that

[TABLE]

Due to the definition of $K$, we deduce that (G1) holds true with $c_{1}=ac_{R}\theta_{0}^{n_{1}}/d$ and the probability measure $\nu(dx)=\frac{\psi(x)}{a\eta(x)}\mathbbm{1}_{K}(x)(\nu_{R}R_{n^{\prime\prime}_{1}})(dx)$.

Defining $\psi_{1}=\eta V_{0}$, we also deduce from (2.4) that,

[TABLE]

In view of the definition of $V_{0}(x)$ for all $x\in E^{\prime}$, we have

[TABLE]

which also makes sense for $x\in E\setminus E^{\prime}$. For such an $x$, we deduce from (2.3) that $P_{n}\psi(x)\leq\zeta_{n}\theta_{0}^{n}\psi(x)$. Without loss of generality, increasing $m$, $\lambda$ and $\rho$ if necessary, we can assume that $\zeta_{m}^{1/m}\leq\lambda<\rho<1$. Then,

[TABLE]

Hence, we have checked that $P_{1}\psi_{1}\leq\theta_{0}\rho\psi_{1}+c_{2}\mathbbm{1}_{K}\psi_{1}$ on $E$ for some constants $\rho<1$ and $c_{2}<+\infty$. Since $P_{1}\eta=\theta_{0}\eta$, the proof of (G2) is completed. Note also that $\psi\leq\psi_{1}$ and the fact that $\psi_{1}\in L^{\infty}(\psi)$ follows from the inequality $P_{n}\psi_{1}\leq A_{n}\psi_{1}$ for some constant $A_{n}$, which is an immediate consequence of (2.3) and the fact that $\eta\in L^{\infty}(\psi_{1})$.

Thanks to Remark 1, it is sufficient to check (G3) with $\psi_{2}=\eta$ instead of $\psi_{1}$. Since $\eta$ is an eigenfunction of $P_{1}$, (G3) is trivial.

Since $K\subset E^{\prime}$, it follows from (2.3) that, for all $x\in K$, $\theta_{0}^{-n}P_{n}(\mathbbm{1}_{K}\psi_{1})(x)$ converges as $n\rightarrow+\infty$ to $\eta(x)\nu_{P}(\mathbbm{1}_{K}\psi_{1})>0$. Hence (G4) is clear. ∎

For continuous time semigroups $(P_{t})_{t\in[0,+\infty)}$, the conclusions of Theorem 2.1 can be easily deduced from properties on the discrete skeleton $(P_{nt_{0}})_{n\in\mathbb{N}}$ (similar properties where already observed in Theorem 5 of [31] and in [10]). In the following result, the function $\eta$ and the positive measure $\nu_{P}$ are the one of Theorem 2.1 applied to the discrete skeleton $(P_{nt_{0}})_{n\in\mathbb{N}}$.

Corollary 2.3**.**

Let $(P_{t})_{t\in[0,+\infty)}$ be a continuous time semigroup. Assume that there exists $t_{0}>0$ such that $(P_{nt_{0}})_{n\in\mathbb{N}}$ satisfies Assumption (G), $\left(\frac{P_{t}\psi_{1}}{\psi_{1}}\right)_{t\in[0,t_{0}]}$ is upper bounded by a constant $\bar{c}>0$ and $\left(\frac{P_{t}\psi_{2}}{\psi_{2}}\right)_{t\in[0,t_{0}]}$ is lower bounded by a constant $\underline{c}>0$. Then there exist some constants $C^{\prime\prime}>0$ and $\gamma>0$ such that, for all measurable functions $f:E\rightarrow\mathbb{R}$ satisfying $|f|\leq\psi_{1}$ and all positive measure $\mu$ on $E$ such that $\mu(\psi_{1})<+\infty$ and $\mu(\psi_{2})>0$,

[TABLE]

In addition, there exists $\lambda_{0}\in\mathbb{R}$ such that $\nu_{P}P_{t}=e^{\lambda_{0}t}\nu_{P}$ for all $t\geq 0$, and $e^{-\lambda_{0}t}P_{t}\psi_{1}$ converges uniformly and exponentially toward $\eta$ in $L^{\infty}(\psi_{1})$ when $t\rightarrow+\infty$. Moreover, there exist some constants $C^{\prime\prime\prime}>0$ and $\gamma^{\prime}>0$ such that, for all measurable functions $f:E\rightarrow\mathbb{R}$ satisfying $|f|\leq\psi_{1}$ and all positive measures $\mu$ on $E$ such that $\mu(\psi_{1})<+\infty$,

[TABLE]

Remark 2*.*

In [2], a similar result is obtained, but with the additional assumptions that $\psi_{2}>0$ on $E$ and $n_{1}=1$. In this restricted case, one can check using Remark 1 that their assumptions are equivalent to ours. The fact that $\psi_{2}$ can vanish allows to deal with non-irreducible semigroups (see [10, Section 6]).

Remark 3*.*

The adaptation of the counterpart of Theorem 2.2 to the countinuous-time setting is straightforward. A similar result was proven in [2], where the authors assume in addition that $\zeta_{n}$ is geometrically decreasing and that $\eta$ is positive.

Proof.

Assuming without loss of generality that $t_{0}=1$ and applying (2.1) to $\mu P_{t}$, where $t\in[0,1]$, and $f$ such that $\mu(\psi_{1})<+\infty$ and $|f|\leq\psi_{1}$, one deduces that

[TABLE]

which implies (2.5). Then, applying this inequality to $\mu=\nu_{P}$ and letting $n$ go to infinity shows that $\nu_{P}P_{t}f/\nu_{P}P_{t}\psi_{1}=\nu_{P}f$ for all $t\geq 0$. Choosing $f=P_{s}\psi_{1}$ entails $\nu_{P}P_{t+s}\psi_{1}=\nu_{P}P_{t}\psi_{1}\,\nu_{P}P_{s}\psi_{1}$ for all $s,t\geq 0$, and hence $\nu_{P}P_{t}\psi_{1}=e^{\lambda_{0}t}\nu_{P}\psi_{1}$ for all $t\geq 0$ for some constant $\lambda_{0}\in\mathbb{R}$ (note that $\theta_{0}=e^{\lambda_{0}}$).

Similarly, inequality (2.2) applied to $\mu=\delta_{x}P_{t}$ and $f=\psi_{1}$ on the one hand and to $\mu=\delta_{x}$ and $f=P_{t}\psi_{1}$ on the other hand implies that $P_{t}\eta(x)=\eta(x)\nu_{P}(P_{t}\psi_{1})=e^{\lambda_{0}t}\eta(x)$ for all $t\geq 0$. Applying again (2.2) to $\mu=\delta_{x}P_{t}$ entails that

[TABLE]

In particular, for all $t\geq 0$,

[TABLE]

and $e^{-\lambda_{0}t}P_{t}\psi_{1}$ converges geometrically to $\eta$ in $L^{\infty}(\psi_{1})$. This concludes the proof of Corollary 2.3 ∎

3 Some applications

Given a positive semigroup $P$ acting on measurable functions on $E$, one can try to directly check Assumption (G) by finding appropriate functions $\psi_{1}$ and $\psi_{2}$. Another natural and equivalent strategy is to find a function $\psi$ such that the semigroup defined by $Q_{n}f=\frac{P_{n}(\psi f)}{c^{n}\psi}$ is sub-Markovian and check that it satisfies Assumption (E) of [10]. The main advantage of this last approach is that $Q$ has a probabilistic interpretation as the semigroup of a sub-Markov process. As such, one can apply all the criteria developed in the above mentioned reference and, more generally, use the intuitions and toolboxes of the theory of stochastic processes. Since both approaches are equivalent, this is rather a question of taste.

In Subsection 3.1, we consider the case of a penalized perturbed dynamical system and check Assumption (G) directly. In subsection 3.2, we consider the case of a penalized diffusion processes and check Assumption (E).

3.1 Perturbed dynamical systems

Let $F:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be a locally bounded measurable function and consider the perturbed dynamical system $X_{n+1}=F(X_{n})+\xi_{n}$ with $(\xi_{i})_{i\in\mathbb{Z}_{+}}$ i.i.d. non-degenerate Gaussian random variables. We are interested in the asymptotic behaviour of the associated Feynman-Kac semigroup

[TABLE]

where $E$ is a measurable subset of $\mathbb{R}^{d}$ with positive Lebesgue measure and $G:E\rightarrow(0,+\infty)$ is a measurable function.

Proposition 3.1**.**

Assume that $1/G$ is locally bounded, $G(x)\leq C\,\exp(|x|)$ for all $x\in E$ and some constant $C>0$ and there exists $p>1$ such that $|x|-p|F(x)|\rightarrow+\infty$ when $|x|\rightarrow+\infty$, then the semigroup $(P_{n})_{n\in\mathbb{N}}$ satisfies Assumption (G).

Proof.

One easily checks that $\psi_{1}(x)=\exp(a|x|)$, where $a>0$ is such that $1/a<p-1$, satisfies

[TABLE]

where $C^{\prime}=C\mathbb{E}e^{(1+a)|\xi_{1}|}$. Now, assume without loss of generality that $B(0,1)\cap E$ has positive Lebesgue measure and set $\theta_{2}:=\inf_{x\in B(0,1)\cap E}P_{1}\mathbbm{1}_{B(0,1)\cap E}(x)/2$, which is clearly positive. It then follows from Markov’s property that

[TABLE]

when $n\rightarrow+\infty$. One easily deduces that, for all $R\geq 1$, $\theta_{2}^{-n}\inf_{x\in B(0,R)\cap E}P_{n}\mathbbm{1}_{B(0,1)\cap E}(x)\rightarrow+\infty$, and hence that $\theta_{2}^{-n}\inf_{x\in B(0,R)\cap E}P_{n}\mathbbm{1}_{B(0,R)\cap E}(x)\rightarrow+\infty$ when $n\rightarrow+\infty$.

We set $\theta_{1}=\theta_{2}/2$ and fix $R\geq 1$ large enough so that $C^{\prime}e^{-a(|x|-p|F(x)|)}\leq\theta_{1}$ for all $|x|\geq R$. It then follows from (3.1) that $P_{1}\psi_{1}\leq\theta_{1}\psi_{1}+c_{2}\mathbbm{1}_{K}\psi_{1}$, where $K:=B(0,R)\cap E$. Setting $\psi_{2}(x)=\sum_{k=0}^{n_{0}}\theta_{2}^{-k}P_{k}\mathbbm{1}_{K}(x)$, we deduce that, for all $x\in E$,

[TABLE]

for $n_{0}$ chosen large enough. Since in addition $P_{k}\mathbbm{1}_{K}\leq P_{k}\psi_{1}\leq(\theta_{1}+c_{2})^{k}\psi_{1}$, $\psi_{2}\in L^{\infty}(\psi_{1})$ and, for all $x\in K$, $\psi_{2}(x)\geq 1\geq e^{-aR}\psi_{1}(x)$. Hence, dividing $\psi_{2}$ by $\|\psi_{2}/\psi_{1}\|_{\infty}$ ends the proof of (G2).

In order to prove (G1), (G3) and (G4), we follow similar arguments as for [10, Prop. 7.2]. Since the adaptation of these arguments is a bit tricky because the function $\psi_{1}$ needs to be taken into account appropriately, we give the details below.

The first step consists in proving that there exists a constant $c>0$ such that, for all measurable $A\subset K$, for all $x\in E$ and all $y\in K$,

[TABLE]

This implies easily (G1) for $n_{1}=1$ and (G4) then follows directly from (G1) (since $n_{1}=1$).

To prove (3.2), we observe that (recall that $A\subset K=E\cap B(0,R)$)

[TABLE]

Because $\xi_{1}$ is a non-degenerate gaussian random variable, it is not hard to check that there exists a constant $C_{R}$ depending only on $R$ (and not on $x\in E$ and $y\in K$) such that $\mathbb{P}(F(x)+\xi_{1}\in E\cap A\cap B(0,R))\leq C_{R}\mathbb{P}(F(y)+\xi_{1}\in E\cap A\cap B(0,R))$. Therefore,

[TABLE]

where $c=C_{R}e^{aR}\sup_{|z|\leq R}G(z)\psi_{1}(z)/\inf_{|z|\leq R}G(z)$. Hence (3.2) is proved.

Next, we observe that the Markov property combined with (G2) implies that, for all $x\in E$ and all $n\geq 1$,

[TABLE]

We also have the property that there exists a constant $c^{\prime}>0$ such that, for all $y\in K$ and all $0\leq k\leq n$,

[TABLE]

As observed in Remark 1, since we already proved (G2), the last property is equivalent to the same one with $\psi_{2}$ instead of $\psi_{1}$. Since $P_{1}\psi_{2}\geq\theta_{2}\psi_{2}$ on $K$ (3.4) is then clear.

The proof of (G3) can then be done by combining the last inequalities. We first decompose $P_{n}\psi_{1}$ depending on the value of the first return time in $K$: for all $x\in E$,

[TABLE]

where we used (3.3) and Markov’s property in the second line. We then proceed by using (3.2) for some fixed $y\in K$ first, (3.3) next, and finally (3.4) twice:

[TABLE]

Since the last factor is bounded in $n$, this ends the proof of Proposition 3.1. ∎

3.2 Diffusion processes

Let $(X_{t})_{t\in[0,+\infty)}$ be solution to the SDE

[TABLE]

where $B=(B^{(1)},\ldots,B^{(d)})$ is a standard $d$-dimensional Brownian motion and $b:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is locally Hölder. Let $r:(0,+\infty)^{d}\rightarrow\mathbb{R}$ be locally bounded and consider the semigroup $(P_{t})_{t\in[0,+\infty)}$ defined by

[TABLE]

The term $\mathbbm{1}_{X_{s}\in(0,+\infty)^{d},\ \forall s\in[0,t]}$ above corresponds to a killing at the boundary of the domain $(0,+\infty)^{d}$. Note that the solution to (3.5) may explode in finite time if $b$ does not satisfy the linear growth condition. However, we assume by convention that $X_{t}\not\in(0,+\infty)^{d}$ after the explosion time, so that (3.6) makes sense. We refer to [10, Sections 4.1 and 12.1] for the precise construction of the process.

One motivation for the study of this semigroup comes from the Feynam-Kac formula. Indeed, when the coefficients are smooth enough (see for instance [30, Section 1.3.3]), this semigroup is solution to the Cauchy linear parabolic partial differential equation

[TABLE]

where ${\cal L}$ is the differential operator of second order

[TABLE]

with Dirichlet boundary conditions.

Proposition 3.2**.**

Assume that

[TABLE]

Then the semigroup $(P_{t})_{t\in[0,+\infty)}$ satisfies the assumptions of Corollary 2.3.

Proof.

We first observe that, setting $\psi(x)=\exp\left(\sum_{i=1}^{d}x_{i}\right)$ and $a:=d/2+\sup_{x\in(0,\infty)^{d}}r(x)+\sum_{i=1}^{d}b_{i}(x)$, we have, for all $x\in(0,+\infty)$,

[TABLE]

Using Girsanov’s theorem, we deduce that

[TABLE]

where $\kappa(y)=a-r(y)-\frac{d}{2}-\sum_{i=1}^{d}b_{i}(y)\geq 0$ and $\bar{X}=(\bar{X}^{(1)},\ldots,\bar{X}^{(d)})$ is solution to the SDE ${\rm d}\bar{X}^{(i)}_{t}={\rm d}B^{(i)}_{t}+(1+b_{i}(\bar{X}_{t}))\,{\rm d}t$ with $\bar{X}^{(i)}_{0}=x_{i}$.

Assumption (3.7) thus implies that the conditions of [10, Theorem 4.5] are satified111To prove (4.12) therein, one can use the same argument as the one used in Corollary 4.3 of this reference. and hence that $Q$ satisfies Assumption (F) therein, which implies that Assumption (E) is satisfied by the semigroup $Q_{nt_{0}}$ for some $t_{0}>0$ and some Lyapunov functions $\varphi_{1}$ and $\varphi_{2}$, that $\left(\frac{Q_{t}\varphi_{1}}{\varphi_{1}}\right)_{t\in[0,t_{0}]}$ is uniformly bounded, and that there exist a positive function $\eta_{Q}\in L^{\infty}(\varphi_{1})$ and a constant $\lambda_{0}>0$ such that $Q_{t}\eta_{Q}=e^{-\lambda_{0}t}\eta_{Q}$ for all $t\in[0,+\infty)$.

To conclude, it remains to observe that the same procedure as the one used in the proof of Theorem 2.1 above allows to deduce from these properties that $(P_{nt_{0}})_{n\geq 0}$ satisfies Assumption (G) with $\psi_{1}=\psi\varphi_{1}$ and $\psi_{2}=\psi\eta_{Q}$. Observing also that $\psi_{2}$ is the function $\eta$ of Theorem 2.1, we deduce that $(P_{t})_{t\in[0,+\infty)}$ satisfies the assumptions of Corollary 2.3. ∎