Infinite-dimensional stochastic differential equations and tail $ \sigma$-fields

Abstract

We present general theorems solving the long-standing problem of the existence and pathwise uniqueness of strong solutions of infinite-dimensional stochastic differential equations (ISDEs) called interacting Brownian motions. These ISDEs describe the dynamics of infinite-many Brownian particles moving in $ \mathbb{R}^d $ with free potential $ \Phi $ and mutual interaction potential $ \Psi $. We apply the theorems to essentially all interaction potentials of Ruelle's class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to logarithmic potentials appearing in random matrix theory. We solve ISDEs of the Ginibre interacting Brownian motion and the sine$_{\beta}$ interacting Brownian motion with $ \beta = 1,2,4$. We also use the theorems in separate papers for the Airy and Bessel interacting Brownian motions. One of the critical points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail $ \sigma $-fields of labeled path spaces consisting of trajectories of infinitely many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions.