Interlacing Diffusions

TL;DR

This paper explores intertwinings of diffusions and their duals, constructing coupled processes and Markovian Gelfand-Tsetlin patterns, with explicit formulas for transition densities and new related processes.

Contribution

It introduces a general framework for interlacing diffusions and their duals, providing explicit formulas and constructing new Markov processes in Gelfand-Tsetlin patterns.

Findings

Couplings interlacing diffusions and their duals.

Explicit formulas for transition densities of coupled processes.

Construction of new processes related to matrix minors.

Abstract

We study in some generality intertwinings between $h$-transforms of Karlin-McGregor semigroups associated with one dimensional diffusion processes and those of their Siegmund duals. We obtain couplings so that the corresponding processes are interlaced and furthermore give formulae in terms of block determinants for the transition densities of these coupled processes. This allows us to build diffusion processes in the space of Gelfand-Tsetlin patterns so that the evolution of each level is Markovian. We show how known examples naturally fit into this framework and construct new processes related to minors of matrix valued diffusions. We also provide explicit formulae for the transition densities of the particle systems with one-sided collisions at either edge of such patterns.