Homogenization of stable-like operators with random, ergodic coefficients

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 $q_{\epsilon }(x,\xi ;\theta )=\frac{1}{{\epsilon }^{\alpha }}q(x\mathrm{/}\epsilon ,\epsilon \xi ;\theta )$, where$$q(x,\xi ;\theta )={\int }_{{\mathbb{R}}^d}(1-e^{iy\cdot \xi }+iy\cdot \xi \text{mathds}{1}_{\{\mathrm{\mid }y\mathrm{\mid }\le 1\}})\frac{⟨a(x;\theta )y,y⟩}{\mathrm{\mid }y{\mathrm{\mid }}^{d+2+\alpha }}dy,$$for $(x,\xi ,\theta )\in {\mathbb{R}}^{2d}\times \mathrm{\Theta }$. 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 $(\mathrm{\Theta },\mathcal{H},m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e.\ $\mathrm{\mid }a(x;\theta )\mathrm{\mid }\le \mathrm{\Lambda }$ and $\text{Tr}(a(x;\theta ))\ge \lambda$ for some $\mathrm{\Lambda },\lambda >0$ and all $\theta \in \mathrm{\Theta }$. In addition, we suppose that the field is regular enough so that for any $\theta \in \mathrm{\Theta }$, 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 $\epsilon \to 0^{+}$, in the Skorokhod space, $m$-a.s.\ in $\theta$, to an $\alpha$-stable process whose Fourier symbol $\bar{q}(\xi )$ is given by $\bar{q}(\xi )={\int }_{\mathrm{\Omega }}q(0,\xi ;\theta ){\mathrm{\Phi }}_{\ast }(\theta )m(d\theta )$, where ${\mathrm{\Phi }}_{\ast }$ is a strictly positive density w.r.t.\ measure $m$. Our result has an analytic interpretation in terms of the convergence, as $\epsilon \to 0^{+}$, of the solutions to random integro-differential equations ${\mathrm{\partial }}_t{u}_{\epsilon }(t,x;\theta )=-q_{\epsilon }(x,D;\theta )u_{\epsilon }(t,x;\theta )$, with the initial condition $u_{\epsilon }(0,x;\theta )=f(x)$, where $f$ is a bounded and continuous function.