Upper bound on heat kernels of finite particle systems of Keller-Segel type

We obtain an upper bound on the heat kernel of the Keller-Segel finite particle system that exhibits blow up effects. The proof exploits a connection between Keller-Segel finite particles and certain non-local operators. The latter allows to address some aspects of the critical behaviour of the Keller-Segel system resulting from its two-dimensionality.