Determinantal Martingales and Noncolliding Diffusion Processes

Abstract

Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in the sense that any spatio-temporal correlation function can be expressed by a determinant. The purpose of the present paper is to clarify the connection between these two aspects. We introduce a notion of determinantal martingale and prove that, if the system has determinantal-martingale representation, then it is determinantal. In order to demonstrate the direct connection between the two aspects, we study three processes.

v6: AMS-LaTeX, 50 pages, no figure, minor corrections made for publication in Stoch. Proc. Appl

References in corpus (8)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Orthogonal polynomial ensembles in probability theory
• Differential Equations for Dyson Processes
• Nonintersecting Brownian excursions
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Zeros of Airy Function and Relaxation Process
• Noncolliding Squared Bessel Processes
• Nonintersecting lattice paths on the cylinder

Cited by in corpus (9)

• Elliptic Determinantal Process of Type A
• Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes
• Elliptic Determinantal Processes and Elliptic Dyson Models
• Partial Isometries, Duality, and Determinantal Point Processes
• Exact solution of interacting particle systems related to random matrices
• Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes
• Determinantal Martingales and Correlations of Noncolliding Random Walks
• Determinantal Point Processes, Stochastic Log-Gases, and Beyond
• Captive Jump Processes