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STOCHASTIC ANALYSIS OF INFINITE PARTICLE SYSTEMS — A NEW DEVELOPMENT IN CLASSICAL STOCHASTIC ANALYSIS AND DYNAMICAL UNIVERSALITY OF RANDOM MATRICES

We examine the stochastic dynamics of infinite particle systems moving in Rd. The classical stochastic analysis pursues the stochastic dynamics of a single particle. We explain how to extend the classical stochastic analysis when the object becomes an infinite particle system. The equilibrium states of a system of infinite number of particles resulting from random matrices have the logarithmic function as an interaction potential. Hence, we develop a theory that applies to interaction potentials having a dominating influence over long distances. As an application, we show that the universality of point processes related to random matrices holds for stochastic dynamics. 1. Beginnings of the general theory of interacting Brownian motion Consider an infinite number of Brownian particles moving in Euclidean space R with interacting potential Ψ:R→R∪{∞}. We denote the position of the particles at time t by Xt = (X i t)i∈N ∈ (Rd)N. Then Xt is given by the following stochastic differential (integral) equation: X t −X 0 = B t − β 2 ∫ t