Strong Markov property of determinantal processes with extended kernels

TL;DR

This paper proves the strong Markov property for limit determinantal processes derived from noncolliding Brownian motion and Bessel processes, and characterizes their associated Dirichlet forms and stochastic differential equations.

Contribution

It establishes the strong Markov property for infinite-particle determinantal processes with extended kernels and identifies their Dirichlet forms and SDEs.

Findings

Proves strong Markov property for limit processes

Identifies Dirichlet forms for these processes

Derives associated infinite-dimensional SDEs

Abstract

Noncolliding Brownian motion (Dyson's Brownian motion model with parameter $\beta=2$) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.