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Journal Articles Bernoulli Year : 2020

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

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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.
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Dates and versions

hal-01673332 , version 1 (29-12-2017)
hal-01673332 , version 2 (19-01-2018)
hal-01673332 , version 3 (13-02-2018)
hal-01673332 , version 4 (03-09-2018)
hal-01673332 , version 5 (06-09-2018)
hal-01673332 , version 6 (25-07-2019)

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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⟩
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