Free Positive Multiplicative Brownian Motion and the Free Additive Convolution of Semicircle and Uniform Distributions

Abstract The free positive multiplicative Brownian motion $$(h_t)_{t\ge 0}$$ ( h t ) t ≥ 0 is the large N limit in non-commutative distribution of matrix geometric Brownian motion. One key property of $$(h_t)_{t\ge 0}$$ ( h t ) t ≥ 0 is the fact that the corresponding spectral distributions $$(\nu _t)_{t\ge 0}$$ ( ν t ) t ≥ 0 form a semigroup with respect to free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $$\nu _t$$ ν t can be expressed by the pushforward measure of a free additive convolution of the semicircle and the uniform semicircle 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 $$\nu _t$$ ν t which generalize the corresponding known moment formulas involving Laguerre polynomials.