Fluctuations for fully pushed stochastic fronts

TL;DR

This paper analyzes the asymptotic fluctuations of stochastic travelling wave solutions in reaction-diffusion equations with Wright-Fisher noise, confirming the fully pushed regime exhibits Brownian motion-like shifts.

Contribution

It provides the first full fluctuation theorem for fully pushed stochastic reaction-diffusion equations, verifying a physical conjecture and developing an infinite-dimensional perturbation method.

Findings

Solutions are asymptotically close to a stochastic shift of the deterministic wave.

The limiting shift process is characterized as a Brownian motion with drift.

The fully pushed regime exhibits fluctuation behavior akin to Brownian motion.

Abstract

We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.