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

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