Martingale Solution to a Stochastic Chemotaxis System with Porous Medium Diffusion

TL;DR

This paper establishes the existence of a martingale solution for a stochastic chemotaxis system with porous medium diffusion, using a fixed point approach in a stochastic setting.

Contribution

It introduces a novel stochastic framework for the Keller-Segel system with porous medium diffusion and constructs a martingale solution via a specialized fixed point theorem.

Findings

Existence of a martingale solution for the stochastic chemotaxis system.

Development of a stochastic fixed point theorem tailored to the problem.

Analysis of the solution's properties in a Banach space setting.

Abstract

In this paper, we study the classical Keller - Segel system on a two-dimensional domain perturbed by a pair of Wiener processes, where the leading diffusion term is replaced by a porous media term. Since the randomness is intrinsic, the interpretation of the stochastic integral in the Stratonovich sense is natural. We construct a solution (integral) operator and establish its continuity and compactness properties in an appropriately chosen Banach space. In this manner, we formulate a stochastic version of the Schauder - Tychonoff Type Fixed Point Theorem which is specific to our problem to obtain a solution. In-kind, we achieve the existence of a martingale solution.