Uniqueness of the stochastic Keller-Segel model in one dimension

TL;DR

This paper proves the pathwise uniqueness and existence of strong solutions for the stochastic Keller-Segel model in one dimension, building on previous existence results and enhancing understanding of solution regularity.

Contribution

It establishes pathwise uniqueness and strong solution existence for the 1D stochastic Keller-Segel system, extending prior existence results and improving solution regularity insights.

Findings

Proved pathwise uniqueness of solutions.

Established existence of strong solutions.

Enhanced regularity results for solutions.

Abstract

In a recent paper (J. Differential Equations, 310: 506-554, 2022), the authors proved the existence of martingale solutions to a stochastic version of the classical Patlak-Keller-Segel system in 1 dimension (1D), driven by time-homogeneous spatial Wiener processes. The current paper is a continuation and consists of two results about the stochastic Patlak-Keller-Segel system in 1D. First, we establish some additional regularity results of the solutions. The additional regularity is, e.g. important for its numerical modeling. Then, as a second result, we obtain the pathwise uniqueness of the solutions to the stochastic Patlak-Keller-Segel system in 1D. Finally, we conclude the paper with the existence of the strong solution to this system in 1D.