Small Random Perturbations of a Dynamical System with Blow-up

TL;DR

This paper investigates how small random noise affects a reaction-diffusion system that can blow up, showing that noise causes blow-up with probability one and analyzing the timing and distribution of this blow-up.

Contribution

It provides a rigorous analysis of the blow-up behavior under stochastic perturbations, including convergence of explosion times and metastability in different initial conditions.

Findings

Perturbed system blows up with probability one

Explosion time converges to deterministic time for certain initial data

System exhibits metastable behavior in the attraction domain of equilibrium

Abstract

We study small random perturbations by additive white-noise of a spatial discretization of a reaction-diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.