Approximation for random stable   manifolds under multiplicative correlated noises

Tao  Jiang; Xianming  Liu; Jinqiao Duan

doi:10.3934/dcdsb.2016091

Article Contents

2016, Volume 21, Issue 9: 3163-3174. Doi: 10.3934/dcdsb.2016091

This issue Previous Article Taylor schemes for rough differential equations and fractional diffusions Next Article Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process

Approximation for random stable manifolds under multiplicative correlated noises

1.
School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan 430074
2.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received: November 30, 2015

Revised: January 31, 2016

Published: October 2016

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Abstract

The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.

Keywords:

Wong-Zakai approximation,
random invariant manifolds,
stochastic evolutionary equations,
correlated noise,
integrated Ornstein-Uhlenbeck processes.

Mathematics Subject Classification: Primary: 37L55, 35R60; Secondary: 60H15, 58J65.

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