  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.
