A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case

TL;DR

This paper introduces a novel stochastic interpretation of the one-dimensional Keller-Segel model using a unique McKean-Vlasov interaction kernel, establishing well-posedness for the system and the associated stochastic process.

Contribution

It presents an original McKean-Vlasov framework with a singular interaction kernel for the Keller-Segel system, extending understanding of its stochastic representation.

Findings

Proved well-posedness of the Keller-Segel system in 1D.

Established existence and uniqueness of the McKean-Vlasov process.

Linked the PDE system to a stochastic particle interpretation.

Abstract

In this paper we analyze a stochastic interpretation of the one-dimensional parabolic-parabolic Keller-Segel system without cut-off. It involves an original type of McKean-Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean-Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed for any values of the parameters of the model.