Multi-scale analysis of SPDEs with degenerate additive noise

Mohammed, Wael W.; Blömker, Dirk; Klepel, Konrad

doi:10.1007/s00028-013-0213-3

Cited by 21 publications

(9 citation statements)

References 20 publications

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“…To give a meaning to problem (1.1), we adapt the concept of local mild solution as in [21]. Definition 2.2.…”

Section: Setting and Assumptionsmentioning

confidence: 99%

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

Fu¹,

Blömker

2019

Preprint

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This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation.Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern.This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple onedimensional stochastic Ginzburg-Landau equation.

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“…To give a meaning to problem (1.1), we adapt the concept of local mild solution as in [21]. Definition 2.2.…”

Section: Setting and Assumptionsmentioning

confidence: 99%

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

Fu¹,

Blömker

2019

Preprint

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“…Recently, Equation (1.1) with r = 2 was studied analytically by [22,23] in the deterministic case, i.e without noise. While in the stochastic case this equation with r = 2 was addressed by [24,25,26,27]. Moreover, several numerical and analytical methods have recently been suggested to solve the space fractional partial di¤erential Equation (1.1) without noise see for instance [28,29,30,31].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of the Space-Fractional Diffusion Equations with Additive Noise

Ww¹,

Ahmad²

2021

Preprint

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In this article we take into account a class of stochastic space diffusion equations with polynomials forced by additive noise. We derive rigorously limiting equations which de…ne the critical dynamics. Also, we approximate solutions of stochastic fractional space di¤usion equations with polynomial term by limiting equations, which are ordinary di¤er-ential equations. Moreover, we address the e¤ect of the noise on the solution’s stabilization. Finally, we apply our results to Fisher’s equation and Ginzburg–Landau models.

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“…Wang and Duan 12 derived an effective equation for a stochastic partial differential equation model under a fast random dynamical boundary condition by reducing the random dynamical boundary condition to a simpler one. See also previous studies 1,15‐22 for related work.…”

Section: Introductionmentioning

confidence: 99%

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

Chen

Lei

2021

Math Methods in App Sciences

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This work is concerned with a class of stochastic partial differential equations with a fast random dynamical boundary condition. In the limit of fast diffusion, it derives an effective stochastic partial differential equation to describe the evolution of the dominant pattern. Using the multiscale analysis and the averaging principle, it then establishes deviation estimates of the original stochastic system towards the effective approximating system. A concrete example further illustrates the result on a large time scale.

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Multi-scale analysis of SPDEs with degenerate additive noise

Cited by 21 publications

References 20 publications

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

Approximate Solutions of the Space-Fractional Diffusion Equations with Additive Noise

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

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