An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes
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.
Cite
@article{arxiv.2602.17495,
title = {An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes},
author = {Andrea Di Primio and Marvin Fritz and Luca Scarpa and Margherita Zanella},
journal= {arXiv preprint arXiv:2602.17495},
year = {2026}
}