Convergence and center manifolds for differential equations driven by colored noise

Abstract

In this paper, we study the convergence and pathwise dynamics of random differential equations driven by colored noise. We first show that the solutions of the random differential equations driven by colored noise with a nonlinear diffusion term uniformly converge in mean square to the solutions of the corresponding Stratonovich stochastic differential equation as the correlation time of colored noise approaches zero. Then, we construct random center manifolds for such random differential equations and prove that these manifolds converge to the random center manifolds of the corresponding Stratonovich equation when the noise is linear and multiplicative as the correlation time approaches zero.

Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise

“…where K 0 (τ, ω) is given by (23). By using (3) and (13), one can easily check that K 0 is tempered, which along with (32) completes the proof.…”

“…The approximate equation can generate a random dynamical system (or cocycle) for a wide class of nonlinearity, which is in sharp contrast with the original SPDEs. The solutions, random attractors, invariant manifolds and foliations of the approximate equation converge to the original one in some sense [15,17,41,13,23,14,21,24,11,22,20,36].…”

“…A very natural question is about the dynamical behavior. Recently, the idea of the Wong-Zakai approximations has been used to investigate the dynamics of 2830 XIAOHU WANG, DINGSHI LI AND JUN SHEN * stochastic equations, including random attractors, invariant manifolds and foliations for stochastic differential equations, see, e.g., [15,11,17,41,13,23,14,21,24,22,20,36]. There are two processess are used to approximate the Brownian motion: Euler approximation and colored noise.…”

“…For the details of Euler approximation please see Section 2. The colored noise is given by The colored noise can be considered as an approximation of the Brownian motion, see Lemma 2.2 of [15,13], also [23].…”

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains

“…The stochastic equation with colored noise can generate a random dynamical system for a wide class of nonlinearities. The solutions and the random dynamical behavior of the equation with colored niose converge to these of the stochastic equation driven by Brownian motion in some sense [6,16,20]. Colored noise can been seen as an approximation for the Brownian motion.…”