# Metastability and exit problems for systems of stochastic reaction-diffusion equations

## TL;DR

This paper develops a metastability theory for stochastic reaction-diffusion equations with small multiplicative noise, analyzing exit times and shapes from complex domains of attraction without relying on finite-dimensional approximations.

## Contribution

It introduces a purely infinite-dimensional approach to metastability in stochastic reaction-diffusion systems with non-gradient nonlinearities and non-uniform attraction domains.

## Key findings

- Proves large deviations asymptotics for exit times.
- Characterizes the most probable exit shapes.
- Addresses systems with saddles and limit cycles in attraction domains.

## Abstract

In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.