Homogenization of random parabolic operator with large potential.
type: Item
instance of: scholarly article
title: Homogenization of random parabolic operator with large potential.
zbMATH Open document ID: 1099.35009
DOI: 10.1016/S0304-4149(00)00095-8
author: F. Campillo
author: Marina Kleptsyna
author: A. Piatnitski
published in: Stochastic Processes and their Applications
publication date: 2004-11-26T00:00:00Z
review text: The homogenization problem for a random parabolic operator with a large potential and with coefficients rapidly oscillating both in time and space is investigated. Namely, a Cauchy problem \[ u^ {\varepsilon }_ {t}(t,x) = \operatorname {div}\left [ a\Bigl ({x\over \varepsilon },\xi (\varepsilon ^ {-\alpha }t)\Bigr ) \nabla u^ {\varepsilon }(t,x)\right ] + \frac 1{\varepsilon ^ {1\land \alpha /2}}c\Bigl ({x\over \varepsilon },\xi (\varepsilon ^ {-\alpha }t\Bigr )u^ \varepsilon (t,x) \tag{1} \] with an initial condition \(u^ \varepsilon (0,x) = u_ 0(x)\), \(x\in \mathbb R^ {m}\), \(t\in [0,T]\), is considered. It is supposed that \(\alpha >0\), \(a(\cdot ,y)\), \(c(\cdot ,y)\) are periodic functions, and \(\xi \) is a stationary ergodic diffusion process in \(\mathbb R^ {d}\) defined by a stochastic differential equation \(\text d\xi (t) = B(\xi (t))\,\text dt + \sigma (\xi (t))\,\text dW(t)\). The behaviour of \(u^ \varepsilon \) as \(\varepsilon \downarrow 0\) is studied under suitable hypotheses on the functions \(a\) and \(c\) and upon mixing properties of the process \(\xi \), the limit problem being strongly dependent on the value of the parameter \(\alpha \). If \(\alpha >2\) then it is shown that \(u^ \varepsilon \) converges in probability to a solution of a Cauchy problem for a non-random parabolic operator with constant coefficients. If \(\alpha \leq 2\) then the law of \(u^ \varepsilon \) converges to a law of a solution to a bilinear stochastic partial differential equation driven by a one-dimensional Wiener process. The limit stochastic PDE is rather different for \(\alpha <2\) and \(\alpha =2\).
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2119490
zbMATH Keywords: random parabolic operators
zbMATH Keywords: homogenization
zbMATH Keywords: averaging
MaRDI profile type: Publication
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full work available at URL: s0304-4149(00)00095-8
OpenAlex ID: W2049140206
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description: scientific article; zbMATH DE number 2119490