"Infinite-dimensional stochastic differential equations and tail Π-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 (IBMs). These ISDEs describe the dynamics of infinite-many Brownian particles moving in Rd with free potential ³ and mutual interaction potential ΅. 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 sineΐ IBMs with ΐ=1,2,4 and the Ginibre IBM. The theorems are also used in separate papers for Airy and Bessel IBMs. One of the key points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail Π-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.
