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

Abstract The free positive multiplicative Brownian motion $$(h_t)_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> is the large N limit in non-commutative distribution of matrix geometric Brownian motion. One key property of $$(h_t)_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> is the fact that the corresponding spectral distributions $$(\nu _t)_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> form a semigroup with respect to free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $$\nu _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> 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 $$\nu _t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> which generalize the corresponding known moment formulas involving Laguerre polynomials.