Mathematics – Probability
Scientific paper
2009-10-21
Mathematics
Probability
14 pages
Scientific paper
We consider an infinite dimensional diffusion on $T^{\mathbb Z^d}$, where $T$ is the circle, defined by an infinitesimal generator of the form $L=\sum_{i\in\mathbb Z^d}(\frac{a_i(\eta)}{2}\partial^2_i +b_i(\eta)\partial_i)$, with $\eta\in T^{\mathbb Z^d}$, where the coefficients $a_i,b_i$ are finite range, bounded with 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 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 an 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 $L^*f_{k+1}=A^*f_k$.
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