Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

• 成果类型: Article
• 署名作者: Chueshov, ID; Vuillermot, PA
• 署名单位: Ministry of Education & Science of Ukraine; VN Karazin Kharkiv National University; Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050186
• 发表日期: 1998
• 页码: 149-202
• 关键词: partial-differential equations; reaction-diffusion equations; almost-periodic attractors; navier-stokes equation; random dynamic-systems; approximation; r(n)

摘要

In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along with a related Wong-Zakai approximation procedure.