Exponential ergodicity and Rayleigh-Schroedinger series for infinite dimensional diffusions

TL;DR

This paper proves exponential ergodicity for infinite-dimensional diffusions on a torus under certain conditions, and develops a Rayleigh-Schroedinger series expansion for perturbed invariant measures, with applications to interaction effects.

Contribution

It establishes exponential ergodicity for a class of infinite-dimensional diffusions and introduces an analytic expansion for invariant measures under perturbations.

Findings

Exponential ergodicity in the uniform norm for the diffusion.

Existence of an analytic expansion for the invariant measure under perturbations.

Quantification of interaction effects on invariant measures.

Abstract

We consider an infinite dimensional diffusion on $T^{\mathbb Z^d}$, where $T$ is the circle, defined by an infinitesimal generator of the form $L=\sum_{i\in\mathbb Z^d}\left(\frac{a_i(\eta)}{2}\partial^2_i +b_i(\eta)\partial_i\right)$, with $\eta\in T^{\mathbb Z^d}$, where the coefficients $a_i,b_i$ are of finite range, bounded with uniformly bounded second order partial derivatives and the ellipticity assumption $\inf_{i,\eta}a_i(\eta)>0$ is satisfied. We prove that whenever $\nu$ is an invariant Gibbs measure for this diffusion satisfying the logarithmic Sobolev inequality, then the dynamics is exponentially ergodic in the uniform norm, and hence $\nu$ is the unique invariant measure. As an application of this result, we prove that if $A=\sum_{i\in\mathbb Z^d}c_i(\eta)\partial_i$, and $c_i$ satisfy the condition $\sum_{i\in\mathbb Z^d} \int c_i^2d\nu<\infty$, then there is an $\epsilon_c>0$, such that for every $\epsilon\in (-\epsilon_c,\epsilon_c)$, the infinite dimensional diffusion with generator $L_\epsilon=L+\epsilon A$, has a unique invariant measure $\nu_\epsilon$ having a Radon-Nikodym derivative $g_\epsilon$ with respect to $\nu$, which admits the analytic expansion $g_\epsilon=\sum_{k=0}^\infty \epsilon^k f_k$, where $f_k\in L_2[\nu]$ are defined through $f_0=1$, $\int f_kd\nu=0$ and the recurrence equations $L^*f_{k+1}=A^*f_k$. We give an example where through this expansion we are able to quantify the effect on the invariant measure of a perturbation triggering interaction on independent diffusions.