Homogenization of random parabolic operator with large potential.
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 \(\alpha =2\).
scientific article; zbMATH DE number 2119490
scientific article