2025/03/25 by Hall, Brian C., Eaknipitsari, Sorawit · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.19682
We consider a family bs,τ of free multiplicative Brownian motions labeled by a real variance parameter s and a complex covariance parameter τ. We then consider the element xbs,τ, where x is non-negative and freely independent of bs,τ. Our goal is to identify the support of the Brown measure of xbs,τ. In the case τ=s, we identify a region Σs such that the Brown measure is vanishing outside of Σs except possibly at the origin. For general values of τ, we construct a map fs-τ and define Ds,τ as the complement of fs-τ(Σsc). Then the Brown measure is zero outside Ds,τ except possibly at the origin. The proof of these results is based on a two-stage PDE analysis, using one PDE (following the work of Driver, Hall, and Kemp) for the case τ=s and a different PDE (following the work of Hall and Ho) to deform the τ=s case to general values of τ.