Interacting diffusions on positive definite matrices

TL;DR

This paper studies systems of Brownian particles on positive definite matrices with simple interactions, revealing integrable structures linked to matrix Bessel functions and multivariate generalizations, connected to quantum Toda chains.

Contribution

It introduces new interacting diffusion processes on positive definite matrices with integrable properties related to advanced special functions.

Findings

Examples of integrable diffusion processes on positive definite matrices.

Connections to matrix Bessel functions and multivariate special functions.

Relation to quantized non-Abelian Toda chain eigenfunctions.

Abstract

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to $K$-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda chain.