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

2022/09/22 by Milica Tomašević, Tomašević, Milica, Guillaume Woessner +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #FOS: Mathematics #Gene Regulatory Network Analysis #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2209.11092

openalex publication_date 2022/09/22 · openalex created_date 2022/09/25 · openalex updated_date 2026/07/28

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 ℝ^d for d≥ 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 ℝ^d. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

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