Escape Rate of Markov Chains on Infinite Graphs

Abstract

We study the escape rate of continuous time symmetric Markov chains associated with weighted graphs. The upper rate functions are given in terms of volume growth of the weighted graphs. For a class of symmetric birth and death processes, we obtain sharp upper rate functions.

Appendix

In this appendix, we collect some basic facts for the stochastic analysis of pure jump martingales and provide a proof for Proposition 6.6. Although results here are not new, it is hard to find explicit references. We include proofs here for the sake of completeness.

The following proposition is a generalization of Theorem 2 in Hamza and Klebaner [28].

Proposition 6.7

Let $(V,w,\mu )$ be a locally finite, connected weighted graph with a fixed reference point $\bar{x}\in V$ and $\left(\left( X_t\right)_{t\ge 0} , \mathcal F ^{\bar{x}}, \left( \mathbb P _{x}\right)_{x\in V_{\infty }}\right)$ be the corresponding minimal càdlàg Markov chain. Assume that the process is stochastically complete. Let $f$ be an unbounded function on $V$ that satisfies the following conditions:

• (1) $f(\bar{x})=0$;

• (2) the sets $\{x\in V: |f(x)|\le n\}$ are finite for all $n\in \mathbb N$;

• (3) $f(x)\ne f(y)$ for any pair $x\sim y$.

Then under $\mathbb P _{\bar{x}}$, the process $M$ defined by

$$\begin{aligned} M_t:= f(X_t)+\int \limits _{0}^{t}(\Delta f)\left( X_s\right)\,\text{ d}s \end{aligned}$$

is a local martingale with respect to the filtration $\mathcal F ^{\bar{x}}$.

Proof

We will follow Jacod and Shiryaev [36] for the theory of random measures and compensators (or dual predictable projections). Every process in the proof will be considered under the probability $\mathbb P _{\bar{x}}$ and the filtration $\left(\mathcal F ^{\bar{x}}_t\right)_{t\ge 0}$. We make the convention that $\inf \emptyset =\infty$. Let in the following

$$\begin{aligned} X_{t-}=\lim _{s\rightarrow t, s<t} X_s, \end{aligned}$$

the existence of which is guaranteed by the càdlàg property of the process.

Consider the process $Y_t:=f(X_t)$ which is a càdlàg pure jump type process on $\mathbb R$. Let $\mu ^f$ be the random measure counting the jumps of $Y$:

$$\begin{aligned} \mu ^f (\omega ; dt, dx):=\sum _{s}\varvec{1}_{\{Y_s(\omega )-Y_{s-}(\omega )\ne 0\}}\delta _{\left( s, Y_s(\omega )-Y_{s-}(\omega )\right)}(dt, dx), \end{aligned}$$

where $\delta _{\left( s,x\right)}$ is the Dirac measure on $\mathbb R \times \mathbb R$ at point $(s, x)$. Note that by the assumption (3) on $f$, $f(X_s)-f(X_{s-})\ne 0$ if and only if $X_s\ne X_{s-}$ (since we have $X_s \sim X_{s-}$ when $X_s\ne X_{s-}$). Define a sequence of stopping times $\left( \tau _n\right)_{n\ge 0}$ by

$$\begin{aligned} \tau _0=0, \tau _{n+1}=\inf \{t>\tau _n: X_t\ne X_{t-}\}, \end{aligned}$$

which are the jumping times of the process $X$. We can rewrite $\mu ^f$ as

$$\begin{aligned} \mu ^f (dt, dx)=\sum _{n\ge 1}\delta _{\left( \tau _n, Y_{\tau _n}-Y_{\tau _n-}\right)}(dt, dx), \end{aligned}$$

where we omit the dependence on the sample $\omega$ for simplicity.

By Theorem 1.33 in [36] and the strong Markov property of the process $X$, we have the compensator of $\mu ^f$ is given by (see, for example [28, 29, 35])

$$\begin{aligned} \nu ^f (dt, \{x\})=\sum _{n\ge 0}\varvec{1}_{\{\tau _n< t\le \tau _{n+1}\}}\text{ deg} \left( X_{\tau _n}\right)\mathbb P _{X_{\tau _n}}[Y_{\tau _{n+1}}-Y_{\tau _{n}}=x] dt. \end{aligned}$$

Let $S_n=\inf \{t>0: |Y_t|>n\}$. Define another sequence of stopping times $\left( T_n\right)_{n\ge 1}$ by $T_n=\min \{n, S_n\}$. Since the process $X$ is stochastically complete, on each finite time interval $[0, t]$, there are only finitely many jumps of $X$, $\mathbb P _{\bar{x}}$ almost surely. It follows that $\mathbb P _{\bar{x}}$ almost surely, for all $t>0$, there is some $n$ such that $S_n>t$, as $f$ is unbounded. Then we see that $\mathbb P _{\bar{x}}$ almost surely $\lim _{n\rightarrow \infty }T_n=\infty .$

Consider the functions $F_n (t, x)=x\cdot \varvec{1}_{\{t\le T_n\}}$. By the property of compensator of a random measure (see for example Theorem 1.8 in [36]), we have for any $n$,

$$\begin{aligned} \nonumber&\mathbb E _{\bar{x}}\left( \int \limits \vert F_n (t,x)\vert \mu ^f (dt, dx)\right)= \mathbb E _{\bar{x}}\left( \int \limits \vert F_n (t,x)\vert \nu ^f (dt, dx)\right) \\&\quad = \mathbb E _{\bar{x}}\left( \int \limits _{0}^{T_n} \frac{1}{\mu (X_{s-})}\sum _{y\in V}w(X_{s-}, y)|f(X_{s-})-f(y)|\,\text{ d}s\right) \end{aligned}$$

(7.1)

$$\begin{aligned}&\quad = \mathbb E _{\bar{x}}\left( \int \limits _{0}^{T_n} \frac{1}{\mu (X_{s})}\sum _{y\in V}w(X_{s}, y)|f(X_{s})-f(y)|\,\text{ d}s\right). \end{aligned}$$

(7.2)

By locally finiteness of $(V,w,\mu )$ and finiteness of the sets $\{x\in V: |f(x)|\le n\}$,

$$\begin{aligned} A_n:= \sup _{\{(t,\omega ): 0<t\le T_n(\omega )\}}\frac{1}{\mu (X_s)}\sum _{y\in V}w(X_s, y)|f(X_s)-f(y)|<\infty . \end{aligned}$$

By (7.2), we have

$$\begin{aligned} \mathbb E _{\bar{x}}\left( \int \limits \vert F_n (t,x)\vert \mu ^f (dt, dx)\right)\le A_n T_n\le n A_n<\infty , \end{aligned}$$

whence the process

$$\begin{aligned} |Y|_t:=\int \limits \varvec{1}_{\{s\le t\}}|x|\mu ^f (dt, dx) \end{aligned}$$

is locally integrable.

Then we have by direct calculation

$$\begin{aligned} Y_t=f(X_t)=f(X_t)-f(X_0)&= \int \limits \varvec{1}_{\{s\le t\}}x\mu ^f (dt, dx),\int \limits _{0}^{t}(\Delta f)\left( X_s\right)\,\text{ d}s \\&= \int \limits _{0}^{t} \frac{1}{\mu (X_s)}\sum _{y\in V}w(X_s, y)\left( f(X_s)-f(y)\right)\,\text{ d}s\nonumber \\&= -\int \limits \varvec{1}_{\{s\le t\}}x\nu ^f (dt, dx) \end{aligned}$$

$\mathbb P _{\bar{x}}$ almost surely, since the right hand side of each equality is well defined. Moreover, by Theorem 1.8 in [36], we have that the process $M$ is a local martingale since

$$\begin{aligned} M_t=Y_t+\int \limits _{0}^{t}(\Delta f)\left( X_s\right) ds=\int \limits \varvec{1}_{\{s\le t\}}x\left( \mu ^f (dt, dx)- \nu ^f (dt, dx)\right). \end{aligned}$$

$\square$

Proposition 7.2

Under the same assumptions of Proposition 7.1, if we assume furthermore that

$$\begin{aligned} \frac{1}{\mu (x)}\sum _{y\in V}w(x,y) (f(x)-f(y))^2\le 1 \end{aligned}$$

(7.3)

for all $x\in V$, then the process $M$ is a martingale, and $\mathbb E _{\bar{x}}(M_t^2)\le t$ for all $t>0$.

Proof

We adopt the same notations as in the proof of Proposition 6.7. The quadratic variation process $[M, M]$ of the local martingale $M$ is given by

$$\begin{aligned}{}[M, M]_t=\sum _{0<s\le t}(f(X_s)-f(X_{s-}))^2=\int \limits \varvec{1}_{\{s\le t\}}x^2 \mu ^f (dt, dx). \end{aligned}$$

Note that the quadratic predictable covariation process $\langle M, M \rangle$ of $M$ satisfies

$$\begin{aligned} \langle M, M \rangle _t=\int \limits \varvec{1}_{\{s\le t\}}x^2 \nu ^f (dt, dx)=\int \limits _{0}^{t} \frac{1}{\mu (X_s)}\sum _{y\in V}w(X_s, y)\left( f(X_s)-f(y)\right)^2\,\text{ d}s. \end{aligned}$$

By (7.3), we have that for all $t>0$,

$$\begin{aligned} \mathbb E _{\bar{x}}\left( [M, M]_t\right)= \mathbb E _{\bar{x}}\left( \langle M, M \rangle _t\right)\le t. \end{aligned}$$

Then it follows from standard results in martingale theory that $M$ is a martingale with

$$\begin{aligned} \mathbb E _{\bar{x}}(M_t^2)=\mathbb E _{\bar{x}}\left( [M, M]_t\right)\le t \end{aligned}$$

for all $t>0$ (see for example Protter [42], Corollary 3, p. 72). $\square$

Remark 7.3

Condition (7.3) corresponds to the adaptedness condition when $f$ is given by the distance from a reference point. From this martingale theory point of view, we also see that the adapted metric is a natural notion.