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

Brian C. Hall, Sorawit Eaknipitsari

TL;DR

This work identifies the support of the Brown measure for xb_{s,τ} with x≥0 freely independent of the free multiplicative Brownian motion b_{s,τ}. It proves that for τ = s the Brown measure vanishes outside the time-dependent domain Σ_s, defined via a lifetime function T(λ) derived from a Hamilton–Jacobi PDE, while for general τ the support is the image-defined domain D_{s,τ} obtained by the holomorphic map f_{s−τ} applied to the complement of Σ_s. The authors develop a two-stage PDE approach: first solve a Driver–Hall–Kemp type PDE at τ = s, then deform to general τ using Hall–Ho PDEs and a deformation map, establishing zero Brown measure outside D_{s,τ} and giving the corresponding boundary behavior and mass at the origin when applicable. This extends prior results on Brown measures with unitary or projection initial data to arbitrary non-negative initial conditions and complex τ, providing a comprehensive PDE-based framework for the support of xb_{s,τ} across the parameter space. The findings have implications for spectral distributions in free probability and random matrix theory, highlighting how non-unitary initial conditions shape the Brown measure’s support via a holomorphic transport mechanism.

Abstract

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 $τ$.

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

TL;DR

This work identifies the support of the Brown measure for xb_{s,τ} with x≥0 freely independent of the free multiplicative Brownian motion b_{s,τ}. It proves that for τ = s the Brown measure vanishes outside the time-dependent domain Σ_s, defined via a lifetime function T(λ) derived from a Hamilton–Jacobi PDE, while for general τ the support is the image-defined domain D_{s,τ} obtained by the holomorphic map f_{s−τ} applied to the complement of Σ_s. The authors develop a two-stage PDE approach: first solve a Driver–Hall–Kemp type PDE at τ = s, then deform to general τ using Hall–Ho PDEs and a deformation map, establishing zero Brown measure outside D_{s,τ} and giving the corresponding boundary behavior and mass at the origin when applicable. This extends prior results on Brown measures with unitary or projection initial data to arbitrary non-negative initial conditions and complex τ, providing a comprehensive PDE-based framework for the support of xb_{s,τ} across the parameter space. The findings have implications for spectral distributions in free probability and random matrix theory, highlighting how non-unitary initial conditions shape the Brown measure’s support via a holomorphic transport mechanism.

Abstract

We consider a family of free multiplicative Brownian motions labeled by a real variance parameter and a complex covariance parameter . We then consider the element , where is non-negative and freely independent of . Our goal is to identify the support of the Brown measure of . In the case , we identify a region such that the Brown measure is vanishing outside of except possibly at the origin. For general values of , we construct a map and define as the complement of . Then the Brown measure is zero outside 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 and a different PDE (following the work of Hall and Ho) to deform the case to general values of .

Paper Structure

This paper contains 18 sections, 34 theorems, 192 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Free multiplicative Brownian motions and their Brown measures
  3. The support of $b_t$ with non-negative initial condition
  4. The case of arbitrary $\tau$
  5. Preliminaries
  6. Free Probability
  7. Free Brownian motions
  8. The Brown measure
  9. The case $\tau =s$
  10. The PDE for the regularized log potential and its solution
  11. The lifetime of the solution and its $\varepsilon_0\rightarrow 0$ limit
  12. The domain and its properties
  13. The "outside the domain" calculation
  14. The general $\tau$ case
  15. Statement of results
  16. ...and 3 more sections

Key Result

Lemma 1.2

Assume that $\lambda_0$ is outside the support of $\mu$. Then in the limit as $\varepsilon_0\rightarrow 0$, we have $\varepsilon(t) \equiv 0$ and $\lambda(t) \equiv \lambda$, for as long as the solution to Hamilton's equations exists.

Figures (8)

  • Figure 1: The domain $\overline{\Sigma}_t$ with the eigenvalues (red dots) of a random matrix approximation to $xb_t$, in the case $\mu =\frac{1}{5}\delta_{1} +\frac{4}{5}\delta_2$
  • Figure 2: The domain $D_{s,\tau}$ along with the eigenvalues (red dots) of a random matrix approximation to $xb_{s,\tau}$, with $s=0.2$ and $\mu =\frac{1}{5}\delta_{1} +\frac{4}{5}\delta_2$.
  • Figure 3: The domain $\overline{\Sigma}_t$ with $\mu = \frac{1}{2}\delta_0 +\frac{1}{2}\delta_1$, for $t=1$ (blue), $t=2$ (orange), and $t=3$ (green).
  • Figure 4: The domain $\overline{\Sigma}_t$ with $\mu =\frac{1}{5}\delta_{\frac{1}{2}} +\frac{1}{5}\delta_1 + \frac{3}{5}\delta_{2}$, for $t =\frac{1}{10}$ (blue), $t=\frac{1}{5}$ (orange), and $t=\frac{1}{2}$ (green).
  • Figure 5: A portion of the domain $\Sigma_t$ with $\mu=\delta_1$ and $t=4.02$
  • ...and 3 more figures

Theorems & Definitions (80)

  • Lemma 1.2
  • Theorem 1.3: Free multiplicative Brownian motion with non-negative initial condition
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 70 more