Existence, Uniqueness, Feller Property and Diricihlet form of the Keller-Segel particle system up to blow up

Abstract

The Keller-Segel particle system as defined in \cite{ft} through the Dirichlet form theory (see \cite{f}) consists of $N$ planar Brownian motions interacting through the kernel $x\in \rr^2 \mapsto θK(x) = -θx/\|x\|^2$, where $θ>0$. One particularity of this model is that collisions (i.e. overlapping of particles) occur depending on the value of $θ$, making the well-posedness of the particle system delicate. In this work, we propose a new notion of solution of this SDE without relying on the Dirichlet form theory and prove uniqueness in law of such solutions. The proof relies on an excursion decomposition of the particle system based on the "barycenter-radius-angle" decomposition in \cite{ft}. As additional results, we prove that the solution is strong Markov, has the Feller property and fits in the definition of solutions in the Dirichlet form sense. Finally we prove that the law of the particle system starting from a configuration where all the particles overlap is explicit.