# Existence of an invariant measure on an infinite dimensional space via Lyapunov functional

Set-up. Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V (x) \leq r \}$ is compact in $\mathcal{X}$ for any $r >0$. Let $(X_t)$ be a Markov process in $\mathcal{X}$ such that $P\{ V(X _t) < \infty \text{ for all } t \geq 0 \} = 1$ and $$V(X _t) - \int ^t _0 g(X _s) ds \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad \quad \quad \quad \quad \quad \quad \quad (1)$$ is a martingale, where $$g(x) \leq c - V(x), \quad \text{ for all } x \in \{z \in \mathcal{X}: V(z) < \infty \} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ for some $c >0$. Furthermore, assume that $(X_t)$ is a Feller process, that is, if $f$ is a bounded continuous function on $\mathcal{X}$, then so is $P_t(\cdot, f)$, where $\{P_t, t \geq 0 \}$ is the family of transition kernels corresponding to the process $(X_t)$.

Under these general conditions, is it possible to deduce the existence of an invariant measure for $(X _t)$?

An attempt to prove the existence of an invariant measure. My idea would be to try to prove that the family of distributions of $\mu _t (\Gamma):=\frac 1T \int _0 ^T P \{X _s \in \Gamma \}ds$ is tight and obtain an invariant measure as a limit point. Imitating the proof of Lemma 9.7 of Chapter 4 of the book by Ethier and Kurtz, we can show that for any $t \geq 1$, $\mu _t (K _r) \to 1$ as $r \to \infty$. Namely, we have

$$0 \leq E V (X _t) = E V (X _0) + E \int ^t _0 g (X _s ) ds \\ = E V (X _0) + E \int ^t _0 g (X _s ) I\{ X_s \in K _r \} ds + E \int ^t _0 g (X _s ) I\{ X_s \notin K _r \} ds \\ \leq E V (X _0) + c t \mu _t (K _r) + (c-r)t[1-\mu _t (K _r)],$$ hence $$\mu _t (K _r) \geq 1 - \frac{c + \frac 1t E V (X _0)}{r}.$$ We see that the family $\{ \mu _n , n \in \mathbb{N} \}$ is tight and thus relatively compact by Prokhorov's theorem, therefore there exists a limit point $\mu$. The measure $\mu$ should be invariant for $(X_t)$ since $(X_t)$ is a Feller process: let $t>0$, $\mu _{T_n} \to \mu$, $T_n \to \infty$, and let $\nu _0 = Law (X _0)$, then for every bounded continuous $f$

$$\int P _t (x,f) \mu (dx) = \lim _n \frac{1}{T_n}\int _0 ^{T_n} ds \int P _t (x,f) P _s (y,dx) \nu _0 (dy) \\ = \lim _n \frac{1}{T_n}\int _0 ^{T_n} ds \int P _{s+t} (y,f) \nu _0 (dy) = \lim _n \frac{1}{T_n}\int _t ^{T_n+t} ds \int P _{s} (y,f) \nu _0 (dy) \\ = \lim _n \Big[ \frac{1}{T_n}\int _0 ^{T_n} + \frac{1}{T_n}\int _{T_n} ^{T_n+t} - \frac{1}{T_n}\int _{0} ^{t} \Big] = \lim _n \langle \mu _{T_n}, f \rangle =\langle \mu , f \rangle$$

Question. Does the proof above work? I am asking because in the mentioned book by Ethier and Kurtz the state space is assumed to be locally compact in corresponding statements, and although it seems for me that in the proof above it is not necessary, I am not sure that I have not missed something. Also, I would appreciate any reference where a similar question is addressed.

Motivation. I would like to prove the existence of an invariant distribution for some interacting particle systems represented by an $\mathbb{N}^{\mathbb{Z^d}}$-valued process, so that $\mathcal{X} = \mathbb{N}^{\mathbb{Z^d}}$, with $V(x) = \sum _{q \in \mathbb{Z} ^d} x(q)s(q)$, where $s$ is some summable positive function.

As far as references for these things, have a look at this webpage for lecture notes by Martin Hairer. In particular, the lectures on "Ergodic properties of Markov processes" are I think exactly what you need (see discussion around Theorem 4.21 therein). The important property for $\mathcal{X}$ is to be a Polish space. Local compactness is not necessary.