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In Talay and Tomasevic [20] we proposed a new stochastic interpretation ofthe parabolic-parabolic Keller-Segel system without cutoff. It involved anoriginal type of McKean-Vlasov interaction kernel which involved all the pasttime marginals of its probability distribution in a singular way. In thepresent paper, we study this McKean-Vlasov representation in thetwo-dimensional case. In this setting there exists a possibility of a blow-upin finite time for the Keller-Segel system if some parameters of the model arelarge. Indeed, we prove the well-posedness of the McKean-Vlasov equation undersome constraints involving a parameter of the model and the initial datum.Under these constraints, we also prove the global existence for theKeller-Segel model in the plane. To obtain this result, we combine PDE analysisand stochastic analysis techniques.

A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The two-dimensional case