Homogenization of stable-like operators with random, ergodic coefficients

Abstract

We show homogenization for a family of ${\mathbb{R}}^d$-valued stable-like processes $(X_t^{\hat{I}\mu ;\hat{I}¸}{)}_{t\ge 0}$, $\hat{I}\mu \in (0,1]$, whose (random) Fourier symbols equal $q_{\hat{I}}\mu (x,\hat{I}¾;\hat{I}¸)=\frac{1}{\hat{I}{\mu }^{\hat{I}}\pm }q(x/\hat{I}\mu ,\hat{I}\mu \hat{I}¾;\hat{I}¸)$, where$$q(x,ξ; θ)=\int_{\mathbb{R}^d}\big(1-e^{i y\cdotξ}+iy\cdotξ\mathds{1}_{\{|y|\le1\}}\big)\,\frac{\langle a(x;θ)y,y\rangle}{|y|^{d+2+α}}\,dy,$$for $(x,\hat{I}¾,\hat{I}¸)\in {\mathbb{R}}^{2d}\times \hat{I}˜$. Here, $\hat{I}\pm \in (0,2)$ and the family $(a(x;\hat{I}¸){)}_{x\in {\mathbb{R}}^d}$ of $d\times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space $(\hat{I}˜,\mathcal{H},m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e.\ $|a(x;\hat{I}¸)|\le \hat{I}›$ and $\text{Tr}(a(x;\hat{I}¸))\ge \hat{I}»$ for some $\hat{I}›,\hat{I}»>0$ and all $\hat{I}¸\in \hat{I}˜$. In addition, we suppose that the field is regular enough so that for any $\hat{I}¸\in \hat{I}˜$, the operator $-q(\cdot ,D;\hat{I}¸)$, defined on the space of compactly supported $C^2$ functions, is closable in the space of continuous functions vanishing at infinity and its closure generates a Feller semigroup. We prove the weak convergence of the laws of $(X_t^{\hat{I}\mu ;\hat{I}¸}{)}_{t\ge 0}$, as $\hat{I}\mu \to 0^{+}$, in the Skorokhod space, $m$-a.s.\ in $\hat{I}¸$, to an $\hat{I}\pm$-stable process whose Fourier symbol $\bar{q}(\hat{I}¾)$ is given by $\bar{q}(\hat{I}¾)={\int }_{\hat{I}}\text{c◯}q(0,\hat{I}¾;\hat{I}¸)\hat{I}{¦}_{\ast }(\hat{I}¸)m(d\hat{I}¸)$, where $\hat{I}{¦}_{\ast }$ is a strictly positive density w.r.t.\ measure $m$. Our result has an analytic interpretation in terms of the convergence, as $\hat{I}\mu \to 0^{+}$, of the solutions to random integro-differential equations ${\partial }_t{u}_{\hat{I}}\mu (t,x;\hat{I}¸)=-q_{\hat{I}}\mu (x,D;\hat{I}¸)u_{\hat{I}}\mu (t,x;\hat{I}¸)$, with the initial condition $u_{\hat{I}}\mu (0,x;\hat{I}¸)=f(x)$, where $f$ is a bounded and continuous function.