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 $xb_{s,\tau }$ , where x is non-negative and freely independent of $b_{s,\tau }$ . Our goal is to identify the support of the Brown measure of $xb_{s,\tau }$ . 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 $ .