Exponential ergodicity and Rayleigh-Schroedinger series for infinite dimensional diffusions

We consider an infinite dimensional diffusion on $T^{\mathbb Z^d}$, where $T$ is the circle, defined by an infinitesimal generator of the form $L=\sum_{i\in\mathbb Z^d}\left(\frac{a_i(η)}{2}\partial^2_i +b_i(η)\partial_i\right)$, with $η\in T^{\mathbb Z^d}$, where the coefficients $a_i,b_i$ are of finite range, bounded with uniformly bounded second order partial derivatives and the ellipticity assumption $\inf_{i,η}a_i(η)>0$ is satisfied. We prove that whenever $ν$ 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 $ν$ is the unique invariant measure. As an application of this result, we prove that if $A=\sum_{i\in\mathbb Z^d}c_i(η)\partial_i$, and $c_i$ satisfy the condition $\sum_{i\in\mathbb Z^d} \int c_i^2dν<\infty$, then there is an $ε_c>0$, such that for every $ε\in (-ε_c,ε_c)$, the infinite dimensional diffusion with generator $L_ε=L+εA$, has a unique invariant measure $ν_ε$ having a Radon-Nikodym derivative $g_ε$ with respect to $ν$, which admits the analytic expansion $g_ε=\sum_{k=0}^\infty ε^k f_k$, where $f_k\in L_2[ν]$ are defined through $f_0=1$, $\int f_kdν=0$ 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.

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