Determinantal Martingales and Noncolliding Diffusion Processes

TL;DR

This paper explores the relationship between harmonic Doob transforms and determinantal structures in noncolliding diffusion processes, introducing determinantal martingales to unify these aspects.

Contribution

It introduces the concept of determinantal martingales and proves their role in characterizing determinantal noncolliding diffusion processes.

Findings

Establishes the link between harmonic Doob transforms and determinantal structures.

Introduces determinantal martingales as a unifying concept.

Analyzes three specific 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.