Support of the Brown measure of a family of free multiplicative Brownian motions with non-negative initial condition

Abstract

We consider a family $b_{s,\tau }$ of free multiplicative Brownian motions labeled by a real variance parameter $s$ and a complex covariance parameter $\tau$. We then consider the element $x{b}_{s,\tau }$, where $x$ is non-negative and freely independent of $b_{s,\tau }$. Our goal is to identify the support of the Brown measure of $x{b}_{s,\tau }$. In the case $\tau =s$, we identify a region ${\Sigma }_s$ such that the Brown measure is vanishing outside of ${\overline{\Sigma }}_s$ except possibly at the origin. For general values of $\tau$, we construct a map $f_{s-\tau }$ and define $D_{s,\tau }$ as the complement of $f_{s-\tau }({\overline{\Sigma }}_s^c)$. Then the Brown measure is zero outside $D_{s,\tau }$ 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 $\tau =s$ and a different PDE (following the work of Hall and Ho) to deform the $\tau =s$ case to general values of $\tau$.

39 pages, 8 figures. A few technical improvements since the first version, notably Proposition 2.7 on the convergence of $S(\lambda ,\epsilon )$ to $s(\lambda )$