Quantitative instability for stochastic scalar reaction-diffusion equations

TL;DR

This paper investigates the instability of stochastic scalar reaction-diffusion equations with multiplicative noise, establishing conditions under which solutions exhibit certain boundedness properties near zero, and advancing the theoretical understanding of their dynamics.

Contribution

It introduces a novel approach to analyze the instability of these equations by constructing Lyapunov functionals and employing stochastic homogenisation techniques.

Findings

Positive Lyapunov exponent implies bounds on negative moments of solutions.

Develops corrector estimates for a Poisson problem in infinite dimensions.

Addresses open problems in the stability analysis of stochastic reaction-diffusion equations.

Abstract

This work studies the instability of stochastic scalar reaction diffusion equations, driven by a multiplicative noise that is white in time and smooth in space, near to zero, which is assumed to be a fixed point for the equation. We prove that if the Lyapunov exponent at zero is positive, then the flow of non-zero solutions admits uniform bounds on small negative moments. The proof builds on ideas from stochastic homogenisation. We require suitable corrector estimates for the solution to a Poisson problem involving an infinite-dimensional projective process, together with a linearisation step that hinges on quantitative parametrix-like arguments. Overall, we are able to construct an appropriate Lyapunov functional for the nonlinear dynamics and address some problems left open in the literature.