Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution

TL;DR

This paper studies the spectral distributions of free positive multiplicative Brownian motion, showing they form a semigroup under free multiplicative convolution and expressing them via free additive convolution of semicircle and uniform distributions, with new proofs and formulas.

Contribution

Provides a new proof for the expression of spectral distributions of free positive multiplicative Brownian motion using moments of free additive convolution.

Findings

Spectral distributions form a semigroup under free multiplicative convolution.

Expressed spectral distributions as images of free additive convolution under exponential map.

Derived new integral formulas generalizing Laguerre polynomial moment formulas.

Abstract

The free positive multiplicative Brownian motion $(h_t)_{t\geq0}$ is the large $N$ limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting $h_t:=g_{t/2}g_{t/2}^*$, where $(g_t)_{t\geq0}$ is a free multiplicative Brownian motion, which is the large $N$ limit in non-commutative distribution of the Brownian motion in $\operatorname{Gl}(N,\mathbb{C})$. One key property of $(h_t)_{t\geq0}$ is the fact that the corresponding spectral distributions $(\nu_t)_{t\geq0}\subset M^1((0,\infty))$ form a semigroup w.r.t. free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $\nu_t$ can be expressed by the image measure of a free additive convolution of the semicircle and the uniform distribution on an interval under the exponential map. In this paper, we provide a new proof of this result by calculating the moments of the free additive convolution of semicircle and uniform distributions on intervals. As a by-product, we also obtain new integral formulas for $\nu_t$ which generalize the corresponding known moment formulas involving Laguerre polynomials.