Infinite-Dimensional Stochastic Differential Equations and Diffusion Dynamics of Coulomb Random Point Fields

Abstract: We consider the stochastic dynamics of infinite particle systems in $\mathbb{R}^d$ interacting through the $d$-dimensional Coulomb potential. For arbitrary inverse temperature $\beta>0$ and all dimensions $d\ge2$, we construct solutions to the associated infinite-dimensional stochastic differential equations (ISDEs). Our main results establish the existence of strong solutions and their pathwise uniqueness. The resulting labeled process is an $(\mathbb{R}^d)^N$-valued diffusion with no invariant measure, while the associated unlabeled process is reversible with respect to the Coulomb random point field. Furthermore, we prove that finite-particle systems converge to these infinite-particle dynamics, thereby providing a rigorous foundation for Coulomb interacting Brownian motion.