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.
"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