Fast transport asymptotics for stochastic RDEs with boundary noise

TL;DR

This paper demonstrates how, under certain conditions, complex stochastic reaction-diffusion equations with boundary noise can be approximated by simpler one-dimensional stochastic differential equations, especially when diffusion dominates reaction, and analyzes the resulting fluctuations.

Contribution

It introduces a method to reduce stochastic reaction-diffusion equations with boundary noise to one-dimensional SDEs under spectral gap conditions, highlighting averaging in fast spatial transport.

Findings

SPDEs can be approximated by 1D SDEs when diffusion is fast.

Spectral gap condition enables averaging in stochastic reaction-diffusion equations.

Analysis of fluctuations around the averaged dynamics.

Abstract

We consider a class of stochastic reaction-diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a suitable one-dimensional stochastic differential equation. This replacement is possible under the assumption of spectral gap for the diffusion and is a result of averaging in the fast spatial transport. We also study the fluctuations around the averaged motion.