The Brown measure of a family of free multiplicative Brownian motions

• 成果类型: Article
• 署名作者: Hall, Brian C.; Ho, Ching-Wei
• 署名单位: University of Notre Dame; Academia Sinica - Taiwan
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01166-5
• 发表日期: 2023
• 页码: 1081-1166
• 关键词: segal-bargmann transform; RANDOM MATRICES; calculus

摘要

We consider a family of free multiplicative Brownian motions b(s,tau) parametrized by a real variance parameter s and a complex covariance parameter tau. We compute the Brown measure mu(s,r) of ub(s,tau) where u is a unitary element freely independent of b(s,tau). We find that mu(s,r) has a simple structure, with a density in logarithmic coordinates that is constant in the tau-direction. These results generalize those of Driver-Hall-Kemp and Ho-Thong for the case tau = s. We also establish a remarkable model deformation phenomenon, stating that all the Brown measures with s fixed and tau varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton-Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver-Hall-Kemp, our methods are substantially different at both the technical and conceptual level.