arXiv2024
We show homogenization for a family of $\mathbb{R}^d$-valued stable-like processes $(X_t^{ε;θ})_{t\ge 0}$, $ε\in(0,1]$, whose (random) Fourier symbols equal $q_ε(x,ξ;θ)=\frac{1}{ε^α}q(x/ε,εξ; θ)$, 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 $(x,ξ,θ)\in\mathbb{R}^{2d}\timesΘ$. Here, $α\in(0,2)$ and the family $(a(x; θ))_{x\in\mathbb{R}^d}$ of $d\times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space $(Θ,{\cal H},m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e.\ $|a(x;θ)|\leΛ$ and $\text{Tr}(a(x;θ))\geλ$ for some $Λ,λ>0$ and all $θ\inΘ$. In addition, we suppose that the field is regular enough so that for any $θ\inΘ$, the operator $-q(\cdot,D;θ)$, 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 $(X_t^{ε;θ})_{t\ge 0}$, as $ε\to0^+$, in the Skorokhod space, $m$-a.s.\ in $θ$, to an $α$-stable process whose Fourier symbol $\bar{q}(ξ)$ is given by $\bar{q}(ξ)=\int_Ωq(0,ξ;θ)Φ_*(θ)\,m(dθ)$, where $Φ_*$ is a strictly positive density w.r.t.\ measure $m$. Our result has an analytic interpretation in terms of the convergence, as $ε\to0^+$, of the solutions to random integro-differential equations $ \partial_tu_ε(t,x;θ)=-q_ε(x,D;θ)u_ε(t,x;θ)$, with the initial condition $u_ε(0,x;θ)=f(x)$, where $f$ is a bounded and continuous function.