In this work, we study a stochastic system of \(N\) particles associated with the parabolic-parabolic Keller-Segel system modelling chemotaxis. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity to the chemical is sufficiently small, we show that this particle system indeed exists for any \(N\geq 2\), we show tightness in \(N\)of its empirical measure, and that any weak limit point of this empirical measure, as \(N\to\infty\), solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.
This is a joint work with N. Fournier (LPSM, SU)