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

Milica Tomasevic (1) , Denis Talay (2, 3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 ASCII - Analyse d’interactions stochastiques intelligentes et coopératives
3 TOSCA - TO Simulate and CAlibrate stochastic models

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.

Mots clés: Chemotaxis model; Singular McKean-Vlasov non-linear stochastic differ- ential equation; Keller-Segel system

Domaines: Probabilités [math.PR]

Milica Tomasevic, Denis Talay. A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case. Bernoulli, 2020, 26 (2), pp.1323-1353. ⟨10.3150/19-BEJ1158⟩. ⟨hal-01673332v6⟩