An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

TL;DR

This paper studies a stochastic Allen-Cahn equation with jump-diffusion noise to model biological damage and repair, proving well-posedness, analyzing long-term behavior, and demonstrating biological phenomena through simulations.

Contribution

It introduces a novel stochastic Allen-Cahn model with jump-diffusion noise for biological damage, proving its well-posedness and analyzing its ergodic properties.

Findings

Model captures damage clustering and repair dynamics.

Proves existence of invariant measures and ergodicity.

Simulation illustrates biological damage phenomena.

Abstract

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.