Wong–Zakai approximations and center manifolds of stochastic differential equations

Shen, Jun; Lu, Kening

doi:10.1016/j.jde.2017.06.005

Cited by 39 publications

(20 citation statements)

References 26 publications

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“…Hairer and Pardoux [16] further used a more general Wong-Zakai correction term instead of infinite Ito-Stratonovich correction term to show pathwise convergence. Various works on Wong-Zakai approximations for stochastic partial differential equations see [15,20,21,32,38,42] and the references therein.…”

Section: Min Yang and Guanggan Chenmentioning

confidence: 99%

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Yang¹,

Guang-gan²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

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Section: Min Yang and Guanggan Chenmentioning

confidence: 99%

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Yang¹,

Guang-gan²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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“…To see this, we replace u δ (s) by u(s) on the right hand side of equation ( 41) and denote it by J c δ (u, ω, ξ). For each u ∈ C γ , ξ ∈ E c , and ω ∈ Ω, using (38), Lemma 3.1, and ( 46), we have that…”

Section: An Elementary Calculation Givesmentioning

confidence: 99%

“…As in section 3, we write the spectrum σ(A) of matrix A as [38]), using the same arguments from in [38] , we have that M c δ (ω) is an random invariant manifold given by the graph of a Lipschitz function for all small δ ≥ 0. Furthermore, using the same procedure as for Theorems 3.1-3.3 in [38], we obtain the following result. For brevity we do not repeat the proof here.…”

Section: ω(T)| ≤ |T|mentioning

confidence: 99%

Convergence and center manifolds for differential equations driven by colored noise

Shen¹,

Lu²,

Wang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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.

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“…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].…”

mentioning

confidence: 96%

“…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.…”

mentioning

confidence: 99%

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

Wang¹,

Li²,

Shen³

2021

DCDS-B

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In this paper, we study the Wong-Zakai approximations given by a stationary process via Euler approximation of Brownian motion and the associated long term behavior of the stochastic wave equation driven by an additive white noise on unbounded domains. We first prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, we show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.

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Wong–Zakai approximations and center manifolds of stochastic differential equations

Cited by 39 publications

References 26 publications

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Convergence and center manifolds for differential equations driven by colored noise

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

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