2025/05/09 by Martin Auer, Auer, Martin · 1 citation
Business, Management and Accounting · Mathematics · #05A19 #33C15 #60E10 #Advanced Queuing Theory Analysis #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Operator Algebras (math.OA) #Primary 46L54 #Probability (math.PR) #Random Matrices and Applications #Secondary 60B20
paper · pdf · doi:10.48550/arxiv.2505.05984
openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The free positive multiplicative Brownian motion (ht)t≥0 is the large N limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting ht:=gt/2gt/2^*, where (gt)t≥0 is a free multiplicative Brownian motion, which is the large N limit in non-commutative distribution of the Brownian motion in Gl(N,ℂ). One key property of (ht)t≥0 is the fact that the corresponding spectral distributions (νt)t≥0⊂ M1((0,∞)) form a semigroup w.r.t. free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that νt 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 νt which generalize the corresponding known moment formulas involving Laguerre polynomials.