# Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise

## TL;DR

This paper develops a framework for analyzing stochastic travelling waves in reaction-diffusion equations with translation-invariant noise, providing long-term approximations for wave profiles and speeds, validated through detailed examples.

## Contribution

It generalizes existing phase tracking methods to SPDEs driven by cylindrical Q-Wiener processes, enabling analysis of stochastic travelling waves with invariant noise.

## Key findings

- Framework for long-term wave profile approximation
- Extension of phase tracking to cylindrical Wiener processes
- Validated approach with two detailed examples

## Abstract

Inspired by applications, we consider reaction-diffusion equations on $\mathbb{R}$ that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [Hamster & Hupkes 2017] and [Hamster & Hupkes 2018] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.