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

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.

References in corpus (11)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• A Wong-Zakai theorem for stochastic PDEs
• An Introduction to Stochastic PDEs
• A mild Ito formula for SPDEs
• On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System
• A Multiscale-Analysis of Stochastic Bistable Reaction-Diffusion Equations
• A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations
• Stability of travelling waves in stochastic Nagumo equations
• Stochastic Integral with respect to Cylindrical Wiener Process
• Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems
• Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

Cited by in corpus (4)

• Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations
• Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise
• Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations
• Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations