Probabilistic Weak Solutions to a Stochastic Chemotaxis System With Porous Medium Diffusion

Source abstract

ABSTRACT In this paper, we study a stochastic variant of the classical Keller–Segel system on a two‐dimensional domain, where the leading diffusion term is replaced by a porous media operator and the dynamics are perturbed by a pair of independent Wiener processes. The model describes the interaction between the cell density and the concentration of a chemoattractant , incorporating nonlinear diffusion, chemotactic sensitivity, production and damping effects, together with multiplicative stochastic perturbations of strengths and . Since the randomness is intrinsic, the stochastic terms are interpreted in the Stratonovich sense. To construct solutions, we introduce an integral operator and establish its continuity and compactness properties in a suitable Banach space. This leads to a stochastic analogue of the Schauder‐Tychonoff‐type fixed point theorem tailored to our framework, which ensures the existence of a martingale solution. Furthermore, we establish pathwise uniqueness, uniqueness in law, and the existence of strong solutions. The uniqueness results, however, require additional assumptions on the chemoattractant noise and the initial condition of .