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

Abstract : 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.
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https://hal.inria.fr/hal-02044262
Contributor : Milica Tomasevic <>
Submitted on : Thursday, August 1, 2019 - 8:07:09 AM
Last modification on : Monday, May 17, 2021 - 9:36:02 AM
Long-term archiving on: : Wednesday, January 8, 2020 - 3:02:26 PM

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

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