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

Milica Tomasevic (1, 2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 TOSCA-POST - Simuler et calibrer des modèles stochastiques

In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time marginals of its probability distribution in a singular way. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the McKean-Vlasov equation under some constraints involving a parameter of the model and the initial datum. Under these constraints, we also prove the global existence for the Keller-Segel model in the plane. To obtain this result, we combine PDE analysis and stochastic analysis techniques.

Mots clés: Keller-Segel system; Singular McKean-Vlasov non-linear stochastic differential equation

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Milica Tomasevic. A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The two-dimensional case. The Annals of Applied Probability, 2021, 31 (1), pp.28. ⟨10.1214/20-AAP1594⟩. ⟨hal-02044262v2⟩