Homogenization of stable-like operators with random, ergodic coefficients

Abstract

We show homogenization for a family of $\mathbb{R}^d$-valued stable-like processes $(X_t^{\epsilon;\theta})_{t\ge 0}$, $\epsilon\in(0,1]$, whose (random) Fourier symbols equal $q_\epsilon(x,\xi;\theta)=\frac{1}{\epsilon^{\alpha}}q(x/\epsilon,\epsilon\xi; \theta)$, where$$q(x,\xi; \theta)=\int_{\mathbb{R}^d}\big(1-e^{i y\cdot\xi}+iy\cdot\xi\mathds{1}_{\{|y|\le1\}}\big)\,\frac{\langle a(x;\theta)y,y\rangle}{|y|^{d+2+\alpha}}\,dy,$$for $(x,\xi,\theta)\in\mathbb{R}^{2d}\times\Theta$. Here, $\alpha\in(0,2)$ and the family $(a(x; \theta))_{x\in\mathbb{R}^d}$ of $d\times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space $(\Theta,{\cal H},m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e.\ $|a(x;\theta)|\le\Lambda$ and $\text{Tr}(a(x;\theta))\ge\lambda$ for some $\Lambda,\lambda>0$ and all $\theta\in\Theta$. In addition, we suppose that the field is regular enough so that for any $\theta\in\Theta$, the operator $-q(\cdot,D;\theta)$, 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 $(X_t^{\epsilon;\theta})_{t\ge 0}$, as $\epsilon\to0^+$, in the Skorokhod space, $m$-a.s.\ in $\theta$, to an $\alpha$-stable process whose Fourier symbol $\bar{q}(\xi)$ is given by $\bar{q}(\xi)=\int_{\Omega}q(0,\xi;\theta)\Phi_*(\theta)\,m(d\theta)$, where $\Phi_*$ is a strictly positive density w.r.t.\ measure $m$. Our result has an analytic interpretation in terms of the convergence, as $\epsilon\to0^+$, of the solutions to random integro-differential equations $ \partial_tu_\epsilon(t,x;\theta)=-q_\epsilon(x,D;\theta)u_\epsilon(t,x;\theta)$, with the initial condition $u_\epsilon(0,x;\theta)=f(x)$, where $f$ is a bounded and continuous function.