The Brown measure of the free multiplicative Brownian motion

• 成果类型: Article
• 署名作者: Driver, Bruce K.; Hall, Brian; Kemp, Todd
• 署名单位: University of California System; University of California San Diego; University of Notre Dame
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01142-z
• 发表日期: 2022
• 页码: 209-273
• 关键词: calculus

摘要

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 Sigma(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 Sigma(t), which is strictly positive and real analytic on Sigma(t). This density has a simple form in polar coordinates: W-t (r, theta) = 1/r(2) w(t)(theta), where w(t) is an analytic function determined by the geometry of the region Sigma(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.