On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

TL;DR

This paper establishes the well-posedness of a McKean-Vlasov SDE with singular, memory-dependent interactions in high dimensions, providing a probabilistic interpretation for the Keller-Segel model.

Contribution

It introduces a novel well-posedness result for a McKean-Vlasov SDE with singular, memory-involving interactions and links it to the Keller-Segel system.

Findings

Proved well-posedness of the non-linear martingale problem with singular kernel

Established probabilistic representation for the Keller-Segel system

Derived explicit smallness conditions for model parameters

Abstract

In this work we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in $\mathbb{R}^d$ for $d\geq 3$. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in $\mathbb{R}^d$. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.