Infinite-dimensional stochastic differential equations and tail $ σ$-fields

Hirofumi Osada, Hideki Tanemura

Published 2014-12-30Version 1

We present a novel method to prove 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 particles moving in $ \Rd $ with free potential $ \Phi $ and mutual interacting potential $ \Psi $. We present general theorems for the existence and pathwise uniqueness of strong solutions of ISDEs and apply them to long range interactions such as logarithmic potentials appearing in random matrix theory. As applications, we prove the existence and pathwise uniqueness of strong solutions of ISDEs such as Sine$ _{\beta}$, Airy$_{\beta } $, Bessel$_{\beta } $ interacting Brownian motions with $ \beta = 1,2,4$, and Ginibre interacting Brownian motions which is the typical example in $ \R ^2$. We also apply them to essentially all Ruelle's class interaction potentials such as Lennard-Jones 6-12 potential. We give a new formulation of solutions of ISDEs in terms of tail $ \sigma $-fields of labeled path spaces consisting of trajectry of infinite-many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite-dimentions. Our general theory solves the long standing problem on the uniqueness and existence of strong solutions of interacting Brownian motions in infinite-dimensions with long range interaction potentials.