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Support of the Brown measure of a family of free multiplicative Brownian motions with non-negative initial condition
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 15 December 2025, pp. 1-42
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- Article
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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 $.
