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 $L=\sum_{i\in\mathbb Z^d}\left(\frac{a_i(\eta)}{2}\partial^2_i +b_i(\eta)\partial_i\right)$,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 $A={\sum }_{i\in {\mathbb{Z}}^d}{c}_i(\eta ){\mathrm{\partial }}_i$, and $c_i$ satisfy the condition ${\sum }_{i\in {\mathbb{Z}}^d}\int c_i^{2}d\nu <\mathrm{\infty }$, then there is an ${ϵ}_c>0$, such that for every $\epsilon\in (-\epsilon_c,\epsilon_c)$,theinfinitedimensionaldiffusionwithgenerator$L_\epsilon=L+\epsilon A$,hasauniqueinvariantmeasure$\nu_\epsilon$ having a Radon-Nikodym derivative $g_{ϵ}$ with respect to $\nu$, which admits the analytic expansion $g_{ϵ}={\sum }_{k=0}^{\mathrm{\infty }}{ϵ}^k{f}_k$, where $f_k\in L_2[\nu]$aredefinedthrough$f_0=1$,$\int f_kd\nu=0$ and the recurrence equations $L^{\ast }{f}_{k+1}=A^{\ast }{f}_k$. 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.