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In this work we firstly prove the well-posedness of the non-linear martingaleproblem related to a McKean-Vlasov stochastic differential equation withsingular interaction kernel in $\mathbb{R}^d$ for $d\geq 3$. The particularityof our setting is that the McKean-Vlasov process we study interacts at eachtime with all its past time marginal laws by means of a singular space-timekernel. Secondly, we prove that our stochastic process is a probabilisticinterpretation for the parabolic-parabolic Keller-Segel system in$\mathbb{R}^d$. We thus obtain a well-posedness result to the latter underexplicit smallness condition on the parameters of the model.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model