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

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.