Strong well-posedness of a singular SDE for signed Coulomb particles

TL;DR

This paper proves the existence and uniqueness of strong solutions for a singular SDE system modeling signed Coulomb particles in 2D, addressing collision behavior and singular interaction challenges.

Contribution

It establishes strong well-posedness and characterizes collision events for a Coulomb particle system with singular interactions, extending previous methods.

Findings

Existence and uniqueness of strong global solutions.

Characterization of collision events between particles.

Handling of singular Coulomb interactions in the SDE system.

Abstract

We consider an SDE system for signed Coulomb particles moving in $\mathbb R^2$. Due to the singular Coulomb interaction force, collisions between particles of opposite sign will happen in finite time. Upon collision, the colliding particles are removed from the system. Our main results are the existence and uniqueness of strong global solutions and the characterization of all possible collisions. The challenge of the proofs is to deal with the singularity of the interactions. We overcome this by using scaling invariance of the process and by putting together several tools from [FT25] developed for the similar Keller--Segel particle system.