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BIRS Workshop Lecture Videos

Stochastic dynamics in infinite dimensions related to random matrices Osada, Hirofumi


We talk about stochastic dynamics whose (unlabeled) equilibrium states are point processes appearing in random matrix theory. These dynamics are called interacting Brownian motions (IBMs). We give various examples of IBMs related to random matrices. For example, sine, Airy, Bessel IBMs in one space dimension, and Ginibre IBM and stochastic dynamics related to zero points of planer Gaussian analytic functions (GAF) in two space dimensions. We construct these except GAF as a pathwise unique strong solution of an infinite dimensional stochastic differential equation (ISDE). Our method is analytic, and based on stochastic analysis. We present a sequence of general theorems to solve ISDEs. We 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. If beta-ensembles satisfy some moment bounds of n-particle approximations, then we can immediately obtain stochastic dynamics using the general theory. When beta=2 and d=1, there exists another construction of stochastic dynamics based on space-time correlation functions, called the algebraic construction. Our method yields the same stochastic dynamics obtained by the algebraic construction. Analytic method gives qualitative information. If we would have a time, we will talk about the following as examples. (1) Girsanov-like formula for tagged particles of IBMs: The resulting IBMs are then locally same as Brownian motions. (2) Dynamical rigidity: Random point fields appearing in random matrix theory have various rigidities. We talk their dynamical counter parts and prove that their global behavior is very rigid and different from Brownian motions. We explain how geometric rigidity yields the dynamical rigidity for interacting Brownian motions in infinite dimensions arising from random matrix theory. This talk is based on the joint work with Hideki Tanemura in Chiba University.

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