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