Diffusion approximation for multi-scale stochastic reaction-diffusion equations

2021/01/11 by Longjie Xie, Li Yang, Xie, Longjie +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2101.03917

openalex publication_date 2021/01/11 · openalex created_date 2021/01/18 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the diffusion approximation for singularly perturbed stochastic reaction-diffusion equation with a fast oscillating term. The asymptotic limit for the original system is obtained, where an extra Gaussian term appears. Such a term is explicitly given in terms of the solution of Poisson equation in Hilbert space. Moreover, we also obtain the optimal rate of convergence, and the convergence rate is shown not to depend on the regularity of the coefficients of the original system with respect to the fast variable, which coincides with the intuition since the fast component has been totally homogenized out in the limit equation.

Citations

• Dynamical Theories of Brownian Motion
• On the Poisson equation and diffusion approximation 3
• Equation of motion for a small rigid sphere in a nonuniform flow

Related

• Diffusion approximation for fully coupled stochastic differential equations
• Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions
• Asymptotic behavior of multiscale stochastic partial differential equations
• Weak approximation for stochastic reaction-diffusion equation near sharp interface limit
• Asymptotic behavior of a class of multiple time scales stochastic\n kinetic equations