Density of Brown measure of free circular Brownian motion

2023/07/17 by László Erdős, Erdős, László, Hong Chang Ji +1
Economics, Econometrics and Finance · Mathematics · #46L54 #60B20 #FOS: Mathematics #Functional Analysis (math.FA) #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2307.08626

openalex publication_date 2023/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Brown measure of the free circular Brownian motion, \boldsymbola+√(t)\boldsymbolx, with an arbitrary initial condition \boldsymbola, i.e. \boldsymbola is a general non-normal operator and \boldsymbolx is a circular element *-free from \boldsymbola. We prove that, under a mild assumption on \boldsymbola, the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support -- either (i) sharp cut, i.e. a jump discontinuity along the boundary, or (ii) quadratic decay at certain critical points on the boundary. Our result is in direct analogy with the previously known phenomenon for the spectral density of free semicircular Brownian motion, whose singularities are either a square-root edge or a cubic cusp. We also provide several examples and counterexamples, one of which shows that our assumption on \boldsymbola is necessary.

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