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 $$(h_t)_{t\ge 0}$$