# A Multiscale-Analysis of Stochastic Bistable Reaction-Diffusion Equations

## TL;DR

This paper performs a multiscale analysis of 1D stochastic bistable reaction-diffusion equations, decomposing solutions into traveling waves with random speed and Gaussian fluctuations, with explicit error estimates.

## Contribution

It extends existing results on stochastic neural fields to a new class of stochastic reaction-diffusion equations, providing explicit error bounds and derived equations for wave speed and fluctuations.

## Key findings

- Decomposition of solutions into traveling wave and fluctuations with error estimates
- Derivation of stochastic differential equation for wave speed
- Linear SPDE describing Gaussian fluctuations

## Abstract

A multiscale analysis of 1D stochastic bistable reaction-diffusion equations with additive noise is carried out w.r.t. travelling waves within the variational approach to stochastic partial differential equations. It is shown with explicit error estimates on appropriate function spaces that up to lower order w.r.t. the noise amplitude, the solution can be decomposed into the orthogonal sum of a travelling wave moving with random speed and into Gaussian fluctuations. A stochastic differential equation describing the speed of the travelling wave and a linear stochastic partial differential equation describing the fluctuations are derived in terms of the coefficients. Our results extend corresponding results obtained for stochastic neural field equations to the present class of stochastic dynamics.