A particle system with explosions: law of large numbers for the density of particles and the blow-up time

TL;DR

This paper establishes a strong law of large numbers for the particle density and explosion time in a rescaled particle system with creation, annihilation, and explosion phenomena, linking it to solutions of a semilinear heat equation.

Contribution

It introduces a rigorous law of large numbers for the density and explosion time of a particle system with creation-annihilation dynamics, connecting stochastic processes to PDE solutions.

Findings

Convergence of particle density to PDE solution in supremum norm.

Law of large numbers for explosion time when f(u)=u^p, 1<p≤3.

Identification of the limiting PDE as a semilinear heat equation.

Abstract

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time.