Branching diffusion in inhomogeneous media

TL;DR

This paper studies how branching diffusion processes evolve over time in media with varying properties, revealing different behaviors depending on the branching intensity, including growth and limiting distributions.

Contribution

It provides a detailed analysis of the asymptotic behavior of branching diffusion processes in inhomogeneous media across different regimes.

Findings

Super-critical regime: exponential growth of particles

Sub-critical and critical regimes: finite limiting number of particles

Distribution of the limiting number in sub-critical and critical regimes

Abstract

We investigate the long-time evolution of branching diffusion processes (starting with a finite number of particles) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical regime, we describe the asymptotics of the number of particles in a given domain. In the sub-critical and critical regimes, we show that the limiting number of particles is finite and describe its distribution.