On the singular values of complex matrix Brownian motion with a matrix drift

TL;DR

This paper analyzes the singular values of complex matrix Brownian motion with a drift, showing they form a Markov process with explicit kernels and connections to Bessel diffusions, extending classical results.

Contribution

It generalizes classical results by providing explicit transition kernels for singular values of complex matrix Brownian motion with drift and describes their relation to Bessel diffusions.

Findings

Singular values form a Markov process with explicit transition kernel.

Connections established to squared Bessel diffusions conditioned to never intersect.

Descriptions extended to a class of one-dimensional diffusions.

Abstract

Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let $\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the $N$ eigenvalues of $\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.