Stochastic analysis of infinite particle systems—A new development in classical stochastic analysis and dynamical universality of random matrices

Abstract

We examine the stochastic dynamics of infinite particle systems moving in R d \mathbb {R}^{d} . 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.