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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 $q_\epsilon(x,\xi;\theta)=\frac{1}{\epsilon^{\alpha}}q(x/\epsilon,\epsilon\xi; \theta)$, where$$q(x,\xi; \theta)=\in...

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Bibliographic Details
Published in:	arXiv.org 2024-09
Main Authors:	Klimsiak, Tomasz, Komorowski, Tomasz, Marino, Lorenzo
Format:	Article
Language:	English
Subjects:	Continuity (mathematics) Convergence Differential equations Ergodic processes Fields (mathematics) Homogenization Mathematical analysis Operators (mathematics) Stochastic processes
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Summary:	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.
ISSN:	2331-8422

Summary:	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.
ISSN:	2331-8422