Wong–Zakai Approximations and Long Term Behavior of Stochastic Partial Differential Equations

• Published in: Journal of dynamics and differential equations Vol. 31; no. 3; pp. 1341 - 1371
• Main Authors: Lu, Kening, Wang, Bixiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Attractors (mathematics); Long term; Mathematical analysis; Mathematics; Mathematics and Statistics; Noise; Ordinary Differential Equations; Partial Differential Equations; Stationary processes; White noise; Upper semicontinuity; Random attractor; Secondary 35B41; Wong–Zakai approximation; Primary 35B40; White noise; 37L30; Stochastic equation
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-017-9626-y

In this paper we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic partial differential equations driven by a white noise. We prove that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. When the stochastic partial differential equation is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.