Journal
BERNOULLI
Volume 26, Issue 2, Pages 1323-1353Publisher
INT STATISTICAL INST
DOI: 10.3150/19-BEJ1158
Keywords
chemotaxis model; Keller-Segel system; singular McKean-Vlasov non-linear stochastic differential equation
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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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