Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case
The authors study a class of semilinear parabolic equations subjected to a spatially homogeneous white noise and prove some results concerning the long time behavior of their solutions. In particular they consider SDE in Stratonovich form \[ du=(\text{div} (k\nabla u)+sg(u))dt +g\circ B, u(x,0,\omega)=\phi(x,\omega)\in (u_0,u_1)\subset \mathbb{R},\;s\in \mathbb{R}, \] where \(u\) is a random scalar field defined on \(D\times \mathbb{R}^+\times\Omega, D\subset \mathbb{R}^n\) subjected to the uniform Neumann condition on \(\partial D\) and prove under some assumptions the existence and uniqueness of a random field \(u_\phi\) which satisfies the above properties as a distribution (in PDE sence). Given \(u_0\leq \text{ess inf }\phi(x,\omega)\leq \text{ess sup }\phi(x,\omega)\leq u_1\), they show that \(u_\phi\) stabilizes almost surely for large times in a suitable topology around a spatially homogeneous process. Namely they prove that firstly, if \(s<0\) and \(g'(u_0)>0\), then \(\lim_{t\to\infty} \ln(\| u_\phi(.,t,\omega)-u_0\| _\infty)=sg'(u_0)\) holds a.s., that secondly, if \(s>0\) and \(g'(u_1)<0\), then \(\lim_{t\to\infty}\ln(\| u_\phi(.,t,\omega)-u_1\| _\infty)=sg'(u_1)\) holds a.s., and that thirdly, if \(s=0\), then the following relations hold a.s \[ \lim\inf_{t\to \infty} \text{ess}\inf_{x\in D} u_\phi(x,t,\omega)= \lim\sup_{t\to \infty} \text{ess}\sup_{x\in D}u_\phi(x,t,\omega)=u_0 \] and \[ \lim\sup_{t\to\infty} \text{ess}\inf_{x\in D} u_\phi(x,t,\omega)= \lim\sup_{t\to \infty} \text{ess}\sup_{x\in D}u_\phi(x,t,\omega)=u_1. \] Let in addition \(G:(u_0,u_1)\to \mathbb{R}\) be any primitive of \({1\over g(u)}\) and \(u_0<b(t)\leq a(t)<u_1\) be arbitrary functions for which limits \(a=\lim_{t\to\infty}t^{-{1\over 2}}(G(a(t))-st) \) and \(b= \lim_{t\to \infty}t^{-{1\over 2}}(G(b(t))-st) \) exist. Then \[ \lim_{t\to\infty}P\{b(t)\leq u_\phi(x,t,\omega)\leq a(t)\}=(2\pi)^{1\over 2}\int^a_b \exp[-{y^2\over 2}]dy \] for almost every \(x\in D\).
scientific article; zbMATH DE number 1221942