Homogenization of stable-like operators with random, ergodic coefficients

Abstract

We show homogenization for a family of $R^d$-valued stable-like processes ${\left(X_t^{\epsilon ;\theta }\right)}_{t\ge 0}$, $\epsilon \in (0,1]$, whose (random) Fourier symbols equal $q_{\epsilon }(x,\xi ;\theta )=\frac{1}{{\epsilon }^{\alpha }}q(\frac{x}{\epsilon },\epsilon \xi ;\theta )$, where$q(x,\xi ;\theta )=\underset{R^d}{\int }(1-e^{iz\cdot \xi }+iz\cdot \xi 1_{\left\{\right|z|\le 1\}})\frac{a(x;\theta )z\cdot z}{|z{|}^{d+2+\alpha }}dz,$ for $(x,\xi ,\theta )\in R^{2d}\times \Theta$. Here $\alpha \in (0,2)$ and the family ${\left(a\right(x;\theta \left)\right)}_{x\in R^d}$ of $d\times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space

. We assume that the random field is deterministically bounded and non-degenerate, i.e. $\left|a\right(x;\theta \left)\right|\le \Lambda$ and $\text{Tr}\left(a\right(x;\theta \left)\right)\ge \lambda$ 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 on $R^d$, 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 ${\left(X_t^{\epsilon ;\theta }\right)}_{t\ge 0}$, as $\epsilon \downarrow 0$, in the Skorokhod space, m-a.s. in θ, to an α-stable process whose Fourier symbol $\overline{q}\left(\xi \right)$ is given by $\overline{q}\left(\xi \right)={\int }_{\Theta }q(0,\xi ;\theta ){\Phi }_{⁎}\left(\theta \right)m\left(d\theta \right)$, where ${\Phi }_{⁎}$ is a strictly positive density w.r.t. measure m. Our result has an analytic interpretation in terms of the convergence, as $\epsilon \downarrow 0$, of the solutions to random integro-differential equations ${\partial }_t{u}_{\epsilon }(t,x;\theta )=-q_{\epsilon }(x,D;\theta )u_{\epsilon }(t,x;\theta )$, with the initial condition $u_{\epsilon }(0,x;\theta )=f\left(x\right)$, where f is a bounded and continuous function on $R^d$.