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

被引：4
作者：
Kening Lu
Bixiang Wang
机构：
[1] Brigham Young University,Department of Mathematics
[2] New Mexico Institute of Mining and Technology,Department of Mathematics
来源：
Journal of Dynamics and Differential Equations | 2019年 / 31卷
关键词：
Random attractor; Upper semicontinuity; White noise; Wong–Zakai approximation; Stochastic equation; Primary 35B40; Secondary 35B41; 37L30;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
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.
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页码：1341 / 1371
页数：30
相关论文
共 62 条
[1]  
Bally V(1995)Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations Ann. Probab. 23 178-222
[2]  
Millet A(1988)A convergence result for stochastic partial differential equations Stochastics 24 423-445
[3]  
Sanz-Solé M(1995)Almost sure approximation of Wong–Zakai type for stochastic partial differential equations Stoch. Process. Appl. 55 329-358
[4]  
Brzezniak Z(1994)Attractors for random dynamical systems Probab. Theory Relat. Fields 100 365-393
[5]  
Capinski M(2013)The Stratonovich heat equation: a continuity result and weak approximations Electron. J. Probab. 18 1-34
[6]  
Flandoli F(1996)Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise Stoch. Stoch. Rep. 59 21-45
[7]  
Brzezniak Z(2014)Random attractors for stochastic evolution equations driven by fractional Brownian motion SIAM J. Math. Anal. 46 22812309-654
[8]  
Flandoli F(2016)Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters SIAM J. Appl. Dyn. Syst. 15 625-239
[9]  
Crauel H(1996)Approximation of the stochastic Navier–Stokes equation Mat. Apl. Comput. 15 227-341
[10]  
Flandoli F(2013)Wong–Zakai type convergence in infinite dimensions Electron. J. Probab. 18 34-85
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