Infinite-dimensional stochastic differential equations for Coulomb random point fields

We study the infinite-dimensional stochastic differential equations (ISDEs) of infinite-particle systems associated with Coulomb random point fields. The stochastic dynamics described by these ISDEs are referred to as Coulomb interacting Brownian motions. In all spatial dimensions $d \ge 2$ and for all inverse temperatures $\beta > 0$, we construct the Coulomb interacting Brownian motions. We prove that the ISDEs admit strong solutions and that pathwise uniqueness holds. The resulting labeled dynamics form an $\RdN$-valued diffusion, possibly without an invariant measure, while the corresponding unlabeled process is a reversible diffusion with respect to the underlying Coulomb random point field. Moreover, we identify the infinite-particle stochastic dynamics as the limit in path space of finite-particle systems driven by stochastic differential equations. This identification is achieved through two approximation schemes: finite-domain systems with reflecting boundary conditions and $N$-particle systems. Although the $N$-particle approximation is more fundamental, its justification relies crucially on the finite-domain approximation together with the uniqueness of solutions to the ISDEs. Previously, only the case $d = 2$ and $\beta = 2$, known as the Ginibre interacting Brownian motion, was understood through random matrix theory and determinantal random point fields. Extending this result beyond the determinantal setting has remained a major difficulty. We introduce a new, conceptually clear method based on stochastic analysis of infinite-particle systems with long-range interactions that yields a rigorous construction of Coulomb interacting Brownian motions. A key ingredient is an explicit computation of the logarithmic derivatives of Coulomb random point fields.