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May 2020 A new McKean–Vlasov stochastic interpretation of the parabolic–parabolic Keller–Segel model: The one-dimensional case
Denis Talay, Milica Tomašević
Bernoulli 26(2): 1323-1353 (May 2020). DOI: 10.3150/19-BEJ1158

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.

Citation

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Denis Talay. Milica Tomašević. "A new McKean–Vlasov stochastic interpretation of the parabolic–parabolic Keller–Segel model: The one-dimensional case." Bernoulli 26 (2) 1323 - 1353, May 2020. https://doi.org/10.3150/19-BEJ1158

Information

Received: 1 February 2018; Revised: 1 April 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166565
MathSciNet: MR4058369
Digital Object Identifier: 10.3150/19-BEJ1158

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 2 • May 2020
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