A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear Sensitivity

TL;DR

This paper develops a stochastic PDE model for chemorepulsion with nonlinear sensitivity, proving existence, uniqueness, and long-term statistical properties of solutions, including convergence to an invariant measure with heavy tails.

Contribution

It introduces a novel SPDE model for chemorepulsion with nonlinear sensitivity and establishes key analytical properties such as existence, uniqueness, and ergodicity.

Findings

Existence and uniqueness of global solutions for the SPDE.

The associated semi-group is Markov and has a unique invariant measure.

Invariant measure exhibits heavy tails beyond Gaussian in $L^p$ norms.

Abstract

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We show that for any suitable initial data there exists a pathwise unique, global solution to the SPDE. Furthermore we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a H\"older-Besov space of positive regularity, which the solution law converges to exponentially fast. We also establish tail bounds on the invariant measure that are heavier than Gaussian when measured using any $L^p$ norm.