infinite dimensional扩散的指数遍历性和Rayleigh-Schroedinger级数
Exponential ergodicity and Rayleigh-Schroedinger series for infinite dimensional diffusions
摘要：
本文考虑$T^{\mathbb Z^d}$上的无穷维扩散问题，其中$T$是由$L=\sum_{i\in\mathbb Z^d}\left(\frac{a_i(η)}{2}\partial^2_i +b_i(η)\partial_i\right)$形式的无穷小生成元定义的圆，其中$η\in T^{\mathbb Z^d}$的系数$a_i,b_i$是有限值域的，有一致有界的二阶偏导数，椭圆性假设$\inf_{i,η}a_i(η)>0$满足.我们证明了，只要$ν$是一个不变的吉布斯测度满足对数Sobolev不等式的扩散，那么动力学是指数遍历的一致范数，因此$ν$是唯一的不变测度。作为这一结果的应用，我们证明了如果$A=\sum_{i\in\mathbb Z^d}c_i(η)\partial_i$和$c_i$满足条件$\sum_{i\in\mathbb Z^d} \int c_i^2dν<\infty$，则存在一个$ε_c>0$，使得对于每个$ε\in (-ε_c,ε_c)$，生成元为$L_ε=L+εA$的无穷维扩散具有唯一不变测度$ν_ε$，该测度具有关于$ν$的Radon-Nikodym导数$g_ε$，它允许解析展开$g_ε=\sum_{k=0}^\infty ε^k f_k$，其中$f_k\in L_2[ν]$通过$f_0=1$、$\int f_kdν=0$和递归方程$L^*f_{k+1}=A^*f_k$定义。我们给出了一个例子，通过这种扩展，我们能够量化的扰动触发相互作用的独立扩散的不变测度的影响。
英文摘要：
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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