Martingale Solution to a Stochastic Chemotaxis System with Porous Medium Diffusion

2022/09/26 by Erika Hausenblas, Hausenblas, Erika, Debopriya Mukherjee +3
Mathematics · #35A01 #35K51 #35K87 #35Q92 #60H15 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Morphological variations and asymmetry #Point processes and geometric inequalities #Probability (math.PR) #secondary 35B65

paper · pdf · doi:10.48550/arxiv.2209.12424

openalex publication_date 2022/09/26 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

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.

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