The free positive multiplicative Brownian motion <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(h_t)_{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the large N limit in non-commutative distribution of matrix geometric Brownian motion. One key property of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(h_t)_{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the fact that the corresponding spectral distributions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\nu _t)_{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> form a semigroup with respect to free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\nu _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nu _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> which generalize the corresponding known moment formulas involving Laguerre polynomials.

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

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2026年5月22日

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Martin Auer

摘要

The free positive multiplicative Brownian motion $(h_t)_{t\ge 0}$ ${(h_{t})}_{t \geq 0}$ is the large N limit in non-commutative distribution of matrix geometric Brownian motion. One key property of $(h_t)_{t\ge 0}$ ${(h_{t})}_{t \geq 0}$ is the fact that the corresponding spectral distributions $(\nu _t)_{t\ge 0}$ ${(ν_{t})}_{t \geq 0}$ 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.