Finite Markov chains coupled to general Markov processes and an application to metastability I

TL;DR

This paper develops a method to couple diffusion processes with Markov chains using eigenfunctions, providing explicit formulas and analyzing metastability, especially in the context of small noise perturbations of dynamical systems.

Contribution

It introduces a novel coupling technique between diffusions and Markov chains based on eigenfunctions, with explicit formulas and analysis of metastability phenomena.

Findings

Coupling diffusion processes to Markov chains using eigenfunctions.

Explicit formulas for conditional probabilities in the coupling.

Analysis of metastability in double-well potentials.

Abstract

We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. Applying a Markov mapping theorem, we use the eigenfunctions of the generator to couple this diffusion to a Markov chain whose generator has eigenvalues equal to the eigenvalues of the diffusion generator that are associated with the local minima and establish explicit formulas for conditional probabilities associated with this coupling. The fundamental question then becomes to relate the coupled Markov chain to the approximate Markov chain suggested by the results of Freidlin and Wentzel. In Part II of this work, we provide a complete analysis of this relationship in the special case of a double-well potential in one dimension. More generally, the coupling can be constructed for a general class of Markov processes and any finite set of eigenvalues of the generator.

Equations

$$X_{\epsilon }(t)=X_{\epsilon }(0)-{\int }_0^t\nabla F(X_{\epsilon }(s))ds+\sqrt{2\epsilon }W(t),$$

$$D_j=\{x\in {\mathbb{R}}^d:\underset{t\to \infty }{\lim }{\phi }_x(t)=x_j\}$$

$$P(X_{\epsilon }(t)\in D_j\mid Y_{\epsilon }(t)=j)\to 1$$

$$E^i[{\tau }_j^{Y_{\epsilon }}]\sim E^{x_i}[{\tau }_{B_{\rho }(x_0)}^{X_{\epsilon }}]$$

$${\int }_{E}Afd{\varpi }_k=-{\lambda }_k{\int }_{E}fd{\varpi }_k,$$

$${\varpi }_k(dx)={\eta }_k(x)\varpi (dx),\text{ where }{\eta }_k\in \overline{C}(E),$$

$${\varpi }_k(E)=0.$$

$${\alpha }_i(x)=1+\sum _{k=1}^m{\xi }_i^{(k)}{\eta }_k(x)$$

$$q_{ij}(x)=Q_{ij}\frac{{\alpha }_j(x)}{{\alpha }_i(x)}.$$

$$Bf(x,i)=Af(x,i)+\sum _{j\ne i}{q}_{ij}(x)(f(x,j)-f(x,i)),$$

$$\mathcal{D}(B)=\{f(x,i)=f_1(x)f_2(i):f_1\in \mathcal{D}(A),f_2\in B(E_0)\}$$

$$\alpha (i,\Gamma )={\int }_{\Gamma }{\alpha }_i(x)\varpi (dx),$$

$$f(X(t),Y(t))-f(X(0),Y(0))-{\int }_0^{t}Bf(X(s),Y(s))ds$$

$$Y(t)=k+\sum _{i\ne j}(j-i)N_{ij}\left({\int }_0^t{1}_{\{i\}}(Y(s))q_{ij}(X(s))ds\right)$$

$$\nu (\Gamma \times \{i\})=p_i\alpha (i,\Gamma ),\Gamma \in \mathcal{B}(E),i\in E_0.$$

$$P(X(t)\in \Gamma \mid {\mathcal{F}}_t^Y)=\alpha (Y(t),\Gamma ),$$

$$\tilde{\psi }(i)\equiv {\int }_S\psi (z)\tilde{\alpha }(i,dz)={\int }_E\psi (x,i){\alpha }_i(x)\varpi (dx)=1+\sum _{j\ne i}{Q}_{ij}<\infty ,$$

$$C=\left\{\left({\int }_{S}f(z)\tilde{\alpha }(\cdot ,dz),{\int }_{S}Bf(z)\tilde{\alpha }(\cdot ,dz)\right):f\in \mathcal{D}(B)\right\}\subset {\mathbb{R}}^{m+1}\times {\mathbb{R}}^{m+1}.$$

$$\overline{f}(i)={\int }_{S}f(z)\tilde{\alpha }(i,dz)={\int }_{E}f(x,i)\alpha (i,dx)={\int }_{E}f(x,i){\alpha }_i(x)\varpi (dx).$$

$$C\overline{f}(i)={\int }_{E}Bf(x,i){\alpha }_i(x)\varpi (dx).$$

$$C\overline{f}(i)=-\sum _{k=0}^m{\xi }_i^{(k)}{\lambda }_k{\int }_{E}f(x,i){\eta }_k(dx)+\sum _{j\ne i}{Q}_{ij}{\int }_E{\alpha }_j(x)(f(x,j)-f(x,i))\varpi (dx).$$

$$-\sum _{k=0}^m{\xi }_i^{(k)}{\lambda }_k{\int }_{E}f(x,i){\eta }_k(dx)$$

$$=\sum _{j=0}^m{Q}_{ij}\sum _{k=0}^m{\xi }_j^{(k)}{\int }_{E}f(x,i){\eta }_k(dx)$$

$$=\sum _{j=0}^m{Q}_{ij}{\int }_{E}f(x,i){\alpha }_j(x)\varpi (dx).$$

$$C\overline{f}(i)=Q_{ii}{\int }_{E}f(x,i){\alpha }_i(x)\varpi (dx)+\sum _{j\ne i}{Q}_{ij}{\int }_{S}f(x,j){\alpha }_j(x)\varpi (dx)=\sum _{j=0}^m{Q}_{ij}\overline{f}(j)=Q\overline{f}(i).$$

$$c_1|x{|}^{a_1}-c_2$$

$${\tilde{c}}_1|x{|}^{{\tilde{a}}_1}-{\tilde{c}}_2\le |F(x)|\le {\tilde{c}}_3|x{|}^{{\tilde{a}}_2}+{\tilde{c}}_4,$$

$$F(x)=F(0)+{\int }_0^1\nabla F(sx)\cdot xds,$$

$$|F(x)|\le |F(0)|+|x|(c_3|x{|}^{a_2}+c_4{)}^{1/2},$$

$$T_x=\inf \{t\ge 0:|{\phi }_x(t)|<R\}.$$

$$-C<F({\phi }_x(t))$$

$$=F(x)-{\int }_0^t|\nabla F({\phi }_x(s)){|}^{2}ds$$

Full text

AMS subject classifications: Primary 60J60; secondary 60H10, 60F10, 60J27, 60J28, 34L10

Keywords and phrases: conditional distributions, coupling, eigenfunctions, Freidlin and Wentzell, Markov mapping theorem, Markov processes, metastability

1 Introduction

Fix $\varepsilon>0$ and consider the stochastic process,

[TABLE]

where $F\in C^{3}(\mathbb{R}^{d})$ and $W$ is a standard $d$-dimensional Brownian motion. For the precise assumptions on $F$, see Section 3.1. Let $\varphi$ be the solution to the differential equation $\varphi^{\prime}=-\nabla F(\varphi)$. We will use $\varphi_{x}$ to denote the solution with $\varphi_{x}(0)=x$. The process $X_{\varepsilon}$ is a small-noise perturbation of the deterministic process $\varphi$.

Suppose that $\mathcal{M}=\{x_{0},\ldots,x_{m}\}$ is the set of local minima of the potential function $F$. The points $x_{j}$ are stable points for the process $\varphi$. For $X_{\varepsilon}$, however, they are not stable. The process $X_{\varepsilon}$ will initially gravitate toward one of the $x_{j}$ and move about randomly in a small neighborhood of this point. But after an exponential amount of time, a large fluctuation of the noise term will move the process $X_{\varepsilon}$ out of the domain of attraction of $x_{j}$ and into the domain of attraction of one of the other minima. We say that each point $x_{j}$ is a point of metastability for the process $X_{\varepsilon}$.

If $X$ is a cadlag process in a complete, separable metric space $S$ adapted to a right continuous filtration (assumptions that are immediately satisfied for all processes considered here) and $H$ is either open or closed, then $\tau^{X}_{H}=\inf\{t>0:X(t)\text{ or }X(t-)\in H\}$ is a stopping time (see, for example, [8, Proposition 1.5]). If $x\in S$, let $\tau^{X}_{x}=\tau^{X}_{\{x\}}$. We may sometimes also write $\tau^{X}(H)$, and if the process is understood, we may omit the superscript.

Let

[TABLE]

be the domains of attraction of the local minima. It is well-known (see, for example, [9], [4, Theorem 3.2], [5, Theorems 1.2 and 1.4], and [7]) that as $\varepsilon\to 0$, $\tau^{X_{\varepsilon}}(D_{j}^{c})$ is asymptotically exponentially distributed under $P^{x_{j}}$. It is therefore common to approximate the process $X_{\varepsilon}$ by a continuous time Markov chain on the set $\mathcal{M}$ (or equivalently on $\{0,\ldots,m\}$). In fact, metastability can be defined in terms of convergence, in an appropriate sense, to a continuous time Markov chain. (See the survey article [15] for details.) Beltrán and Landim [2, 3] introduced a general method for proving the metastability of a Markov chain. Along similar lines, Rezakhanlou and Seo [19] developed such a method for diffusions. For an alternative approach using intertwining relations, see [1].

In this project, for each $\varepsilon>0$, we wish to capture this approximate Markov chain behavior by coupling $X_{\varepsilon}$ to a continuous time Markov chain, $Y_{\varepsilon}$, on $\{0,\ldots,m\}$. We will refer to the indexed collection of coupled processes, $\{(X_{\varepsilon},Y_{\varepsilon}):\varepsilon>0\}$, as a coupling sequence. Our objective is to investigate the possibility of constructing a coupling sequence which satisfies

[TABLE]

as $\varepsilon\to 0$, for all $j$. We also want the transition rate for $Y_{\varepsilon}$ to go from $i$ to $j$ to be asymptotically equivalent as $\varepsilon\to 0$ to the transition rate for $X_{\varepsilon}$ to go from a neighborhood of $x_{i}$ to a neighborhood of $x_{j}$. That is, we would like

[TABLE]

as $\varepsilon\to 0$, for all $i$ and $j$, where $B_{\rho}(x)$ is the ball of radius $\rho$ centered at $x$.

In this paper (Part I), we develop our general coupling construction. The construction goes beyond the specific case of interest here. It is a construction that builds a coupling between a Markov process on a complete and separable metric space and a continuous-time Markov chain where the generators of the two processes have common eigenvalues. The coupling is done in such a way that observations of the chain yield quantifiable conditional probabilities about the process. This coupling construction is built in Section 2 and uses the Markov mapping theorem (Theorem A.3). In Section 3, we apply this construction method to reversible diffusions in $\mathbb{R}^{d}$ driven by a potential function with a finite number of local minima.

With this coupling construction in hand, we can build the coupling sequences described above. In our follow-up work (Part II), we take up the question of the existence and uniqueness of a coupling sequence that satisfies requirements (1.3) and (1.4).

2 The general coupling

2.1 Assumptions and definitions

Given a Markov process $X$ with generator $A$ satisfying Assumption 2.1, we will use the Markov mapping theorem to construct a coupled pair, $(X,Y)$, in such a way that for a specified class of initial distributions, $Y$ is a continuous-time Markov chain on a finite state space. The construction then allows us to explicitly compute the conditional distribution of $X$ given observations of $Y$.

For explicit definitions of the notation used here and throughout, see Section A.1.

Assumption 2.1**.**

Let $E$ be a complete and separable metric space.

1. (i)

$A\subset\overline{C}(E)\times\overline{C}(E)$. 2. (ii)

$A$* has a stationary distribution $\varpi\in\mathcal{P}(E)$, which implies $\int_{E}Af\,d\varpi=0$ for all $f\in\mathcal{D}(A)$.* 3. (iii)

For some $m$, there exist signed measures $\varpi_{1},\ldots,\varpi_{m}$ on $E$ and positive real numbers $\lambda_{1},\ldots,\lambda_{m}$ such that, for each $k\in\{1,\ldots,m\}$ and $f\in\mathcal{D}(A)$,

[TABLE]

We define $\varpi_{0}=\varpi$ and $\eta_{0}=1$.

Remark 2.2*.*

If $(1,0)\in A$, then (2.1) implies (2.3).

Remark 2.3*.*

In what follows, we will make use of the assumption that the functions $\eta_{k}$ are continuous. However, this assumption can be relaxed by appealing to the methods in Kurtz and Stockbridge [14].

Assumption 2.4**.**

Let $E$ be a complete and separable metric space. Let $A\subset\overline{C}(E)\times\overline{C}(E)$, $m\in\mathbb{N}$, $Q\in\mathbb{R}^{(m+1)\times(m+1)}$, and $\xi^{(1)},\ldots,\xi^{(m)}\in\mathbb{R}^{m+1}$.

1. (i)

$A$* and $m$ satisfy Assumption 2.1.* 2. (ii)

$Q$* is the generator of a continuous-time Markov chain with state space $E_{0}=\{0,1,\ldots,m\}$ and eigenvalues $\{0,-\lambda_{1},\ldots,-\lambda_{m}\}$.* 3. (iii)

The vectors $\xi^{(1)},\ldots,\xi^{(m)}$ are right eigenvectors of $Q$, corresponding to the eigenvalues $-\lambda_{1},\ldots,-\lambda_{m}$. 4. (iv)

For each $i\in\{0,1,\ldots,m\}$, the function

[TABLE]

satisfies $\alpha_{i}(x)>0$ for all $x\in E$.

We define $\xi^{(0)}=(1,\ldots,1)^{T}$, so that the function $\alpha:E\to\mathbb{R}^{m+1}$ is given by $\alpha=\sum_{k=0}^{m}\xi^{(k)}\eta_{k}$.

Remark 2.5*.*

Given $(A,m,Q)$ satisfying (i) and (ii) of Assumption 2.4, it is always possible to choose vectors $\xi^{(1)},\ldots,\xi^{(m)}$ satisfying (iii) and (iv). This follows from the fact that each $\eta_{k}$ is a bounded function.

Definition 2.6**.**

Suppose $(A,m,Q,\xi^{(0)},\ldots,\xi^{(m)})$ satisfies Assumption 2.4. For $0\leq j\neq i\leq m$, define

[TABLE]

Note that $q_{ij}\in C(E)$. Let $S=E\times E_{0}$. Define $B\subset\overline{C}(S)\times C(S)$ by

[TABLE]

where we take

[TABLE]

In particular, $Af(x,i)=f_{2}(i)Af_{1}(x)$.

For each $i\in E_{0}$, define the measure $\alpha(i,\cdot)$ on $E$ by

[TABLE]

for all $\Gamma\in\mathcal{B}(E)$. Note that by (2.4), (2.3), and (2.2), these are probability measures.

2.2 Construction of the coupling

We are now ready to construct our coupled pair, $(X,Y)$, which will have generator $B$, to prove, for appropriate initial conditions, that the marginal process $Y$ is a Markov chain with generator $Q$, and to establish our conditional probability formulas. We first require two lemmas.

Lemma 2.7**.**

In the setting of Definition 2.6, let $X$ be a cadlag solution of the martingale problem for $A$. Then there exists a cadlag process $Y$ such that $(X,Y)$ solves the (local) martingale problem for $B$. If $X$ is Markov, then $(X,Y)$ is Markov. If the martingale problem for $A$ is well-posed, then the martingale problem for $B$ is well-posed.

Remark 2.8*.*

We are not requiring the $q_{ij}$ to be bounded, so for the process we construct,

[TABLE]

may only be a local martingale.

Proof of Lemma 2.7.

Let $X(t)$ be a cadlag solution to the martingale problem for $A$. Let $\{N_{ij}:i,j\in E_{0},i\neq j\}$ be a family of independent unit rate Poisson processes, which is independent of $X$. Then the equation

[TABLE]

has a unique solution, and as in [12], the process $Z=(X,Y)$ is a solution of the (local) martingale problem for $B$. If $X$ is Markov, the uniqueness of the solution of (2.9) ensures that $(X,Y)$ is Markov. Similarly, $A$ well-posed implies $B$ is well posed. $\Box$

Lemma 2.9**.**

Let $A$ satisfy Assumption 2.1. Taking $\psi(x,i)=1+\sum_{j\neq i}q_{ij}(x)\geq 1$, if $A$ satisfies Condition A.1, then $B$ satisfies Condition A.1 with $E$ replaced by $S=E\times E_{0}$.

Proof.

Since $\mathcal{D}(A)$ is closed under multiplication, $\mathcal{D}(B)$ defined in (2.7) is closed under multiplication.

Since we are assuming that $\mathcal{R}(A)\subset\bar{C}(E)$, for each $f\in\mathcal{D}(B)$, there exists $c_{f}>0$ such that $|Bf(x,i)|\leq c_{f}\psi(x)$.

Condition A.1(iii) for $A$ and the separability of $B(E_{0})$ implies Condition A.1(iii) for $B_{0}$.

Since $A$ is a pre-generator and $B$ is a perturbation of $A$ by a jump operator, $B_{0}$ is a pre-generator. $\Box$

Theorem 2.10**.**

Suppose $A$ satisfies Condition A.1 and $(A,m,Q,\xi^{(1)},\ldots,\xi^{(m)})$ satisfies Assumption 2.4. Let $B$ be given by (2.6) and for $p_{i}\geq 0$, $\sum_{i=0}^{m}p_{i}=1$, define

[TABLE]

If $\widetilde{Y}$ is a cadlag $E_{0}$-valued Markov chain with generator $Q$ and initial distribution $\{p_{i}\}$, then there exists a solution $(X,Y)$ of the martingale problem for $(B,\nu)$ such that $Y$ and $\widetilde{Y}$ have the same distribution on $D_{E_{0}}[0,\infty)$, and

[TABLE]

for all $t\geq 0$ and all $\Gamma\in\mathcal{B}(E)$.

Proof.

We apply Theorem A.3 to the operator $B\subset\overline{C}(S)\times C(S)$.

Let $\gamma:S\to E_{0}$ be the coordinate projection. Let $\widetilde{\alpha}$ be the transition function from $E_{0}$ into $S$ given by the product measure $\widetilde{\alpha}(i,\cdot)=\alpha(i,\cdot)\otimes\delta_{i}^{E_{0}}$, where $\alpha(i,\cdot)$ is given by (2.8). Then $\widetilde{\alpha}(i,\gamma^{-1}(i))=1$ and

[TABLE]

for each $i\in E_{0}$. Define

[TABLE]

The result follow by Theorem A.3, if we can show that $Cv=Qv$ for every vector $v\in\mathcal{D}(C)$. Given $f\in\mathcal{D}(B)$, let

[TABLE]

Note that

[TABLE]

Since $\lambda_{0}=0$, by (2.1) and the definition of $q_{ij}(x)$,

[TABLE]

By assumption $Q\xi^{(k)}=-\lambda_{k}\xi^{(k)}$, so $-\xi_{i}^{(k)}\lambda_{k}=\sum_{j=0}^{m}Q_{ij}\xi_{j}^{(k)}$ and

[TABLE]

This gives

[TABLE]

It follows that $\widetilde{Y}$ is a solution to the martingale problem for $(C,p)$.

By Theorem A.3(a), there exists a solution $Z=(X,Y)$ of the martingale problem for $(B,\nu)$ such that $Y=\gamma(Z)$ and $\widetilde{Y}$ have the same distribution on $D_{E_{0}}[0,\infty)$. Theorem A.3(b) implies (2.10). $\Box$

Remark 2.11*.*

In what follows, we may still write expectations with the notation $E^{x}$ or $E^{i}$, even when we have a coupled process, $(X,Y)$. The meaning will be determined by context, depending on whether the integrand of the expectation involves only $X$ or only $Y$.

3 Reversible diffusions

3.1 Assumptions on the potential function

We now consider the special case of our coupling when $X$ is a reversible diffusion on $\mathbb{R}^{d}$ driven by a potential function $F$ and a small white noise perturbation. We will need to use several results from the literature about the eigenvalues and eigenfunctions of the generator of $X$. We assume the following on $F$.

Assumption 3.1**.**

1. (i)

$F\in C^{3}(\mathbb{R}^{d})$* and $\lim_{|x|\rightarrow\infty}F(x)=\infty$.* 2. (ii)

$F$* has $m+1\geq 2$ local minima ${\cal M}=\{x_{0},\ldots,x_{m}\}$.* 3. (iii)

There exist constants $a_{i}>0$ and $c_{i}>0$ such that $a_{2}<2a_{1}-2$, and

[TABLE]

Remark 3.2*.*

Note that $2<a_{1}\leq a_{2}$. To see this, observe that (3.1) implies $a_{1}\leq a_{2}$. Thus, $a_{1}\leq a_{2}<2a_{1}-2$, which implies $a_{1}>2$.

Lemma 3.3**.**

Under Assumption 3.1, there exist constants $\widetilde{c}_{i}>0$ such that

[TABLE]

where $\widetilde{a}_{i}=a_{i}/2+1$.

Proof.

Since

[TABLE]

it follows from (3.1) that

[TABLE]

and the upper bound in (3.3) follows immediately.

Since $F\to\infty$, there exists $C>0$ such that $F(x)>-C$ for all $x\in\mathbb{R}^{d}$, and since $|\nabla F|\to\infty$, there exists $R>0$ such that $|\nabla F(x)|\geq 1$ whenever $|x|\geq R$.

Recall that $\varphi_{x}$ satisfies $\varphi^{\prime}_{x}=-\nabla F(\varphi_{x})$ and $\varphi_{x}(0)=x$, and define

[TABLE]

Suppose there exists $x$ such that $T_{x}=\infty$. Then, for all $t>0$,

[TABLE]

Therefore, $F(x)\geq t-C$ for all $t$, a contradiction, and we must have $T_{x}<\infty$ for all $x\in\mathbb{R}^{d}$.

Let $L=\sup_{|x|\leq R}F(x)$. By (3.1) and the fact that $F\to\infty$, we may choose $R^{\prime}\geq R$ and $C^{\prime}>0$ such that $F(x)>L$ and $|\nabla F(x)|\geq C^{\prime}|x|^{a_{1}/2}$ whenever $|x|>R^{\prime}$.

Fix $x\in\mathbb{R}^{d}$ with $|x|>2R^{\prime}$, so that $F(x)>L$. Since $|\varphi_{x}(T_{x})|=R$, it follows that $F(\varphi_{x}(T_{x}))\leq L$. By the continuity of $\varphi_{x}$, we may choose $T^{\prime}\in(0,T_{x}]$ such that $F(\varphi_{x}(T^{\prime}))=L$. We then have

[TABLE]

Let $T^{\prime\prime}=\inf\{t\geq 0:|\varphi_{x}(t)|<|x|/2\}$. Note that $F(\varphi_{x}(T^{\prime}))=L$ implies $|\varphi_{x}(T^{\prime})|\leq R^{\prime}<|x|/2$, and therefore $T^{\prime\prime}\leq T^{\prime}$. Moreover, for all $t<T^{\prime\prime}$, we have $|\varphi_{x}(t)|\geq|x|/2>R^{\prime}$, which implies

[TABLE]

Thus,

[TABLE]

But $\int_{0}^{T^{\prime\prime}}|\varphi_{x}^{\prime}(t)|\,dt$ is the length of $\varphi_{x}$ from $t=0$ to $t=T^{\prime\prime}$, which is bounded below by

[TABLE]

Therefore, for all $|x|>2R^{\prime}$, we have $F(x)\geq C^{\prime\prime}|x|^{a_{1}/2+1}-|L|$, where $C^{\prime\prime}=2^{-a_{1}/2-1}C^{\prime}$, and this proves the lower bound in (3.3). $\Box$

3.2 Spectral properties of the generator

Having established our assumptions on $F$, we now turn our attention to the diffusion process, $X_{\varepsilon}$, given by (1.1). To simplify notation, we may sometimes omit the $\varepsilon$. The process $X$ has generator $A=\varepsilon\Delta-\nabla F\cdot\nabla$. To show that $A$ meets the requirements of our coupling from Section 2, we must prove certain results about its eigenvalues and eigenfunctions. For this, we begin with some notation, a lemma, and two results from the literature.

Define $\pi(x)=\pi_{\varepsilon}(x)=e^{-F(x)/2\varepsilon}$. Let

[TABLE]

Lemma 3.4**.**

Let $V_{\varepsilon}$ be given by (3.4), where $F$ satisfies Assumption 3.1. Recall the constants $a_{i}$ from (3.1)-(3.2). For all $\varepsilon\in(0,1)$, there exist constants $c_{i,\varepsilon}>0$ such that

[TABLE]

In particular, $V_{\varepsilon}\to\infty$ for all $\varepsilon\in(0,1)$.

Proof.

Fix $\varepsilon\in(0,1)$. By (3.1) and (3.2), for $x$ sufficiently large,

[TABLE]

and

[TABLE]

for some $0<c\leq C<\infty$. Note that

[TABLE]

Hence, for $x$ sufficiently large, $V_{1}(x)\leq C_{1}|x|^{a_{2}}$. Also,

[TABLE]

Since $a_{1}>a_{2}/2$, it follows that for $x$ sufficiently large, $V_{1}(x)\geq\widetilde{c}|x|^{a_{1}}$. Therefore, there exist constants $\widetilde{c}_{i}>0$ such that

[TABLE]

and

[TABLE]

for all $x\in\mathbb{R}^{d}$. Note that

[TABLE]

so that

[TABLE]

From here, the lemma follows easily. $\Box$

The following two theorems are from [6]. Theorem 3.5 is a consequence of [6, Theorem 4.5.4] and [6, Lemma 4.2.2]. Theorem 3.6 is part of [6, Theorem 2.1.4].

Theorem 3.5**.**

Let $H=-\Delta+W$, where $W$ is continuous with $W\to\infty$. Let $\lambda$ denote the smallest eigenvalue of $H$, and $\psi$ the corresponding eigenfunction, normalized so that $\|\psi\|_{L^{2}(\mathbb{R}^{d})}=1$. Define $Uf=\psi f$ and $\widetilde{H}=U^{-1}(H-\lambda)U$. If

[TABLE]

where $\widehat{a}_{i}>0$, $\widehat{c}_{i}>0$, and $\widehat{a}_{2}<2\widehat{a}_{1}-2$, then $e^{-\widetilde{H}t}$ is an ultracontractive symmetric Markov semigroup on $L^{2}(\mathbb{R}^{d},\psi(x)^{2}\,dx)$. That is, for each $t\geq 0$, the operator $e^{-\widetilde{H}t}$ is a bounded operator mapping $L^{2}(\mathbb{R}^{d},\psi(x)^{2}\,dx)$ to $L^{\infty}(\mathbb{R}^{d},\psi(x)^{2}\,dx)$.

Theorem 3.6**.**

Let $e^{-Ht}$ be an ultracontractive symmetric Markov semigroup on $L^{2}(\Omega,\mu)$, where $\Omega$ is a locally compact, second countable Hausdorff space and $\mu$ is a Borel measure on $\Omega$. If $\mu(\Omega)<\infty$, then each eigenfunction of $H$ belongs to $L^{\infty}(\Omega,\mu)$.

This next proposition establishes the spectral properties of $A$ that are needed to carry out the construction of our coupling.

Proposition 3.7**.**

Fix $\varepsilon>0$. The operator $H=-\Delta+V_{\varepsilon}$ is a self-adjoint operator on $L^{2}(\mathbb{R}^{d})$ with discrete, nonnegative spectrum $\widehat{\lambda}_{k}\uparrow\infty$ and corresponding orthonormal eigenfunctions $\psi_{k}$. Each $\psi_{k}$ is locally Hölder continuous. Moreover, $\widehat{\lambda}_{0}=0$ is simple and $\psi_{0}$ is proportional to $\pi$. We define $\mu$ by $\mu(dx)=\pi(x)^{2}\,dx$ and $\varpi=Z^{-1}\mu$, where $Z=\mu(\mathbb{R}^{d})$. The operator $\widetilde{H}$ given by $\widetilde{H}f=\pi^{-1}H(\pi f)$ is a self-adjoint operator on $L^{2}(\varpi)$ with eigenvalues $\widehat{\lambda}_{k}$ and orthogonal eigenfunctions $\widehat{\eta}_{k}=\psi_{k}/\pi$. The functions $\widehat{\eta}_{k}$ have norm one in $L^{2}(\mu)$, whereas the functions $\eta_{k}=Z^{1/2}\widehat{\eta}_{k}$ have norm one in $L^{2}(\varpi)$.

For $f\in C_{c}^{\infty}(\mathbb{R}^{d})$, we have $-\varepsilon\widetilde{H}f=\varepsilon\Delta f-\nabla F\cdot\nabla f$. Hence, if we define $A$ by

[TABLE]

then $A$ is the generator for the diffusion process given by (1.1). For each $x\in\mathbb{R}^{d}$, (1.1) has a unique, global solution for all time, so that the process $X$ with $X(0)=x$ is a solution to the martingale problem for $(A,\delta_{x})$. The operator $A$ is graph separable, and $\mathcal{D}(A)$ is separating and closed under multiplication. The measure $\varpi$ is a stationary distribution for $A$. Moreover,

[TABLE]

where $\varpi_{k}(dx)=\eta_{k}(x)\varpi(dx)$ and $\lambda_{k}=\varepsilon\widehat{\lambda}_{k}$. The signed measures $\varpi_{k}$ satisfy $\varpi_{k}(\mathbb{R}^{d})=0$, and each $\eta_{k}$ belongs to $\overline{C}(\mathbb{R}^{d})$, the space of bounded, continuous functions on $\mathbb{R}^{d}$.

Proof.

Note that $V\to\infty$ by Lemma 3.4. Therefore, by [18, Theorem XIII.67], we have that $H$ is a self-adjoint operator on $L^{2}(\mathbb{R}^{d})$ with compact resolvent. It follows (see [6, pp. 108–109, 119–120, and Proposition 1.4.3]) that $H$ has a purely discrete spectrum and there exists a complete, orthonormal set of eigenfunctions $\{\psi_{k}\}_{k=0}^{\infty}$ with corresponding eigenvalues $\widehat{\lambda}_{k}\uparrow\infty$. Moreover, $\widehat{\lambda}_{0}$ is simple and $\psi_{0}$ is strictly positive.

Since $V$ is locally bounded, and $(-\Delta+V-\widehat{\lambda}_{k})\psi_{k}=0$, [10, Theorem 8.22] implies that, for each compact $K\subset\mathbb{R}^{d}$, $\psi_{k}$ is Hölder continuous on $K$ with exponent $\gamma(K)$.

Define $U:L^{2}(\mu)\to L^{2}(\mathbb{R}^{d})$ by $Uf=\pi f$, so that $\widetilde{H}=U^{-1}HU$. Since $U$ is an isometry, $\widetilde{H}$ is self-adjoint on $L^{2}(\mu)$ and has the same eigenvalues as $H$. Note that, for any $f\in\mathcal{D}(\widetilde{H})$, it follows from Green’s identity that

[TABLE]

Using the product rule, $\nabla(gh)=g\nabla h+h\nabla g$, this simplifies to

[TABLE]

showing that $\widetilde{H}$ cannot have a negative eigenvalue. Hence, $\widehat{\lambda}_{0}\geq 0$.

By (3.3), we have $\pi\in L^{2}(\mathbb{R}^{d})$, so that $\pi\in\mathcal{D}(H)$ with $H\pi=0$. Hence, since $\widehat{\lambda}_{0}$ is nonnegative and has multiplicity one, it follows that $\widehat{\lambda}_{0}=0$ and $\psi_{0}$ is proportional to $\pi$.

Observe that, if $f\in C_{c}^{\infty}$, then, using the product rule for the Laplacian and the identity $V=\Delta\pi/\pi$, we have

[TABLE]

Since $2\varepsilon\nabla\pi/\pi=-\nabla F$, we have $-\varepsilon\widetilde{H}f=\varepsilon\Delta f-\nabla F\cdot\nabla f$.

Since $\nabla F$ is locally Lipschitz, (1.1) has a unique solution up to an explosion time (see [17, Theorem V.38]). Since $\lim_{|x|\rightarrow\infty}F=\infty$ by assumption and $\lim_{|x|\rightarrow\infty}AF(x)=\infty$ by Lemma 4.2, it follows that $F$ is a Liapunov function for $X_{\varepsilon}$ proving that $X_{\varepsilon}$ does not explode.

By [13, Remark 2.5], $A$ is graph separable. Clearly $\mathcal{D}(A)$ is closed under multiplication. Since $\mathcal{D}(A)$ separates points and $\mathbb{R}^{d}$ is complete and separable, $\mathcal{D}(A)$ is separating (see [8, Theorem 3.4.5]).

If $f\in C_{c}^{\infty}$, then

[TABLE]

so that $\varpi$ is a stationary distribution for $A$. For $k\geq 1$, since $\varpi_{k}(dx)=\eta_{k}(x)\varpi(dx)$, we have

[TABLE]

Also, $\varpi_{k}(\mathbb{R}^{d})=\left\langle{\eta_{k},1}\right\rangle_{L^{2}(\varpi)}=0$, since $\eta_{k}$ and $\eta_{0}=1$ are orthogonal.

Finally, since $\eta_{k}=Z^{1/2}\psi_{k}/\pi$ and $\psi_{k}$ is locally Hölder continuous, it follows that each $\eta_{k}$ belongs to $C(\mathbb{R}^{d})$, and the fact that they are bounded follows from Theorems 3.5 and 3.6. $\Box$

3.3 The coupled process

By Proposition 3.7, the pair $(A,m)$ satisfies Assumption 2.1 with $E=\mathbb{R}^{d}$, so we have the following.

Theorem 3.8**.**

Let $A$ be the generator for (1.1) where $F$ satisfies Assumption 3.1, and let $(-\lambda_{0},\eta_{0}),\ldots,(-\lambda_{m},\eta_{m})$ be the first $m+1$ eigenvalues and eigenvectors of $A$. Let $Q\in\mathbb{R}^{(m+1)\times(m+1)}$ be the generator of a continuous-time Markov chain with state space $E_{0}=\{0,1,\ldots,m\}$ and eigenvalues $\{0,-\lambda_{1},\ldots,-\lambda_{m}\}$ and eigenvectors $\xi^{(1)},\ldots,\xi^{(m)}$ such that $\alpha_{i}$ defined by (2.4) is strictly positive. Let $B$ be defined as in Definition 2.6.

Let $\widetilde{Y}$ be a continuous time Markov chain with generator $Q$ and initial distribution $p=(p_{0},\ldots,p_{m})\in\mathcal{P}(E_{0})$. Then there exists a cadlag Markov process $(X,Y)$ with generator $B$ and initial distribution $\nu$ given by

[TABLE]

such that $Y$ and $\widetilde{Y}$ have the same distribution on $D_{E_{0}}[0,\infty)$, and

[TABLE]

for all $t\geq 0$, all $0\leq j\leq m$, and all $\Gamma\in\mathcal{B}(E)$.

Remark 3.9*.*

That $Q$ with these properties exists can be seen from [16, Theorem 1]. Remark 2.5 ensures the existence of the eigenvectors.

Proof.

Note that under the assumptions of the theorem, $(A,m,Q,\xi^{(1)},\ldots,\xi^{(m)})$ satisfies Assumption 2.4. By Proposition 3.7, the rest of the hypotheses of Theorem 2.10 are also satisfied. Consequently, the process $(X,Y)$ exists, and by uniqueness of the martingale problem for $B$, $(X,Y)$ is Markov. $\Box$

We can now construct the coupling sequences described in the introduction. For each $\varepsilon>0$, choose a matrix $Q_{\varepsilon}$ and eigenvectors $\xi_{\varepsilon}^{(1)},\ldots,\xi_{\varepsilon}^{(m)}$ that satisfy the assumptions of Theorem 3.8. If $(X_{\varepsilon},Y_{\varepsilon})$ is the Markov process described in Theorem 3.8, then the family, $\{(X_{\varepsilon},Y_{\varepsilon}):\varepsilon>0\}$, forms a coupling sequence.

The coupling sequence is determined by the collection, $\{Q_{\varepsilon},\xi_{\varepsilon}^{(1)},\ldots,\xi_{\varepsilon}^{(m)}:\varepsilon>0\}$. By making different choices for the matrices and eigenvectors, we can obtain different coupling sequences. In our follow-up paper, we will consider the question of existence and uniqueness of a coupling sequence that satisfies conditions (1.3) and (1.4).

Acknowledgments

This paper was completed while the first author was visiting the University of California, San Diego with the support of the Charles Lee Powell Foundation. The hospitality of that institution, particularly that of Professor Ruth Williams, was greatly appreciated.

Appendix A Appendix

A.1 The Markov mapping theorem

Let $E$ be a complete and separable metric space, $\mathcal{B}(E)$ the $\sigma$-algebra of Borel subsets of $E$, and $\mathcal{P}(E)$ the family of Borel probability measures on $E$. Let $M(E)$ be the collection of all real-valued, Borel measurable functions on $E$, and $B(E)\subset M(E)$ the Banach space of bounded functions with $\|f\|_{\infty}=\sup_{x\in E}|f(x)|$. Let $\overline{C}(E)\subset B(E)$ be the subspace of bounded continuous functions, while $C(E)$ denotes the collection of continuous, real-valued functions on $E$. A collection of functions $D\subset\overline{C}(E)$ is separating if $\mu,\nu\in\mathcal{P}(E)$ and $\int f\,d\mu=\int f\,d\nu$ for all $f\in D$ implies $\mu=\nu$.

Condition A.1**.**

1. (i)

$B\subset\overline{C}(E)\times C(E)$* and $\mathcal{D}(B)$ is closed under multiplication and separating.* 2. (ii)

There exists $\psi\in C(E)$, $\psi\geq 1$, such that for each $f\in\mathcal{D}(B)$, there exists a constant $c_{f}$ such that

[TABLE]

(We write $Bf$ even though we do not exclude the possibility that $B$ is multivalued. In the multivalued case, each element of $Bf$ must satisfy the inequality.) 3. (iii)

There exists a countable subset $B_{c}\subset B$ such that every solution of the (local) martingale problem for $B_{c}$ is a solution of the (local) martingale problem for $B$. 4. (iv)

$B_{0}f\equiv\psi^{-1}Bf$* is a pre-generator, that is, $B_{0}$ is dissipative and there are sequences of functions $\mu_{n}:E\to\mathcal{P}(E)$ and $\lambda_{n}:E\to[0,\infty)$ such that for each $(f,g)\in B$,*

[TABLE]

for each $x\in E$.

Remark A.2*.*

Condition A.1(iii) holds if $B_{0}$ is graph-separable, that is, there is a countable subset $B_{0,c}$ of $B_{0}$ such that $B_{0}$ is a subset of the bounded, pointwise closure of $B_{0,c}$.

An operator is a pre-generator if for each $x\in E$, there exists a solution of the martingale problem for $(B,\delta_{x})$.

For a measurable $E_{0}$-valued process $Y$, where $E_{0}$ is a complete and separable metric space, let

[TABLE]

Theorem A.3**.**

Let $(S,d)$ and $(E_{0},d_{0})$ be complete, separable metric spaces. Let $B$ satisfy Condition A.1. Let $\gamma:S\to E_{0}$ be measurable, and let $\widetilde{\alpha}$ be a transition function from $E_{0}$ into $S$ (that is, $\widetilde{\alpha}:E_{0}\times\mathcal{B}(S)\to\mathbb{R}$ satisfies $\widetilde{\alpha}(y,\cdot)\in\mathcal{P}(S)$ for all $y\in E_{0}$ and $\widetilde{\alpha}(\cdot,\Gamma)\in B(E_{0})$ for all $\Gamma\in\mathcal{B}(S)$) satisfying $\int h\circ\gamma(z)\,\widetilde{\alpha}(y,dz)=h(y)$, $y\in E_{0}$, $h\in B(E_{0})$, that is, $\widetilde{\alpha}(y,\gamma^{-1}(y))=1$. Assume that $\widetilde{\psi}(y)\equiv\int_{S}\psi(z)\widetilde{\alpha}(y,dz)<\infty$ for each $y\in E_{0}$ and define

[TABLE]

Let $\mu\in\mathcal{P}(E_{0})$ and define $\nu=\int\widetilde{\alpha}(y,\cdot)\,\mu(dy)$.

1. a)

If $\widetilde{Y}$ satisfies $\int_{0}^{t}E[\widetilde{\psi}(\widetilde{Y}(s))]\,ds<\infty$ a.s. for all $t>0$ and $\widetilde{Y}$ is a solution of the martingale problem for $(C,\mu)$, then there exists a solution $Z$ of the martingale problem for $(B,\nu)$ such that $\widetilde{Y}$ has the same distribution on $M_{E_{0}}[0,\infty)$ as $Y=\gamma\circ Z$. If $Y$ and $\widetilde{Y}$ are cadlag, then $Y$ and $\widetilde{Y}$ have the same distribution on $D_{E_{0}}[0,\infty)$. 2. b)

Let ${\bf T}^{Y}=\{t:Y(t)\text{ is }\widehat{\mathcal{F}}^{Y}_{t}\text{ measurable}\}$ (which holds for Lebesgue-almost every $t$). Then for $t\in{\bf T}^{Y}$,

[TABLE] 3. c)

If, in addition, uniqueness holds for the martingale problem for $(B,\nu)$, then uniqueness holds for the $M_{E_{0}}[0,\infty)$-martingale problem for $(C,\mu)$. If $\widetilde{Y}$ has sample paths in $D_{E_{0}}[0,\infty)$, then uniqueness holds for the $D_{E_{0}}[0,\infty)$-martingale problem for $(C,\mu)$. 4. d)

If uniqueness holds for the martingale problem for $(B,\nu)$, then $Y$ restricted to ${\bf T}^{Y}$ is a Markov process.

Remark A.4*.*

If $Y$ is cadlag with no fixed points of discontinuity (that is $Y(t)=Y(t-)$ a.s. for all $t$), then $\widehat{\mathcal{F}}^{Y}_{t}=\mathcal{F}_{t}^{Y}$ for all $t$.

Remark A.5*.*

The main precursor of this Markov mapping theorem is [13, Corollary 3.5]. The result stated here is a special case of Corollary 3.3 of [11].