The Brown measure of the free multiplicative Brownian motion

• Published in: Probability theory and related fields 2022-10, Vol.184 (1-2), p.209-273
• Main Authors: Driver, Bruce K., Hall, Brian, Kemp, Todd
• Format: Article
• Language: English
• Subjects: Analytic functions; Brownian motion; Density; Economics; Eigenvalues; Empirical analysis; Finance; Insurance; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Partial differential equations; Polar coordinates; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Theoretical
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-022-01142-z

The free multiplicative Brownian motion b t is the large- N limit of the Brownian motion on GL ( N ; C ) , in the sense of ∗ -distributions. The natural candidate for the large- N limit of the empirical distribution of eigenvalues is thus the Brown measure of b t . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region Σ t that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density W t on Σ ¯ t , which is strictly positive and real analytic on Σ t . This density has a simple form in polar coordinates: W t ( r , θ ) = 1 r 2 w t ( θ ) , where w t is an analytic function determined by the geometry of the region Σ t . We show also that the spectral measure of free unitary Brownian motion u t is a “shadow” of the Brown measure of b t , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.