The Brown measure of the free multiplicative Brownian motion

Driver, Bruce K.; Hall, Brian C.; Kemp, Todd

doi:10.1007/s00440-022-01142-z

The Brown measure of the free multiplicative Brownian motion

Bruce K. Driver
(
- UC, San Diego
)
,
Brian C. Hall
(
- U. Notre Dame, Dept. Math.
)
,
Todd Kemp
(
- UC, San Diego
)

Mar 26, 2019

64 pages

Published in:

Probab.Theor.Related Fields 184 (2022) 1-2, 209-273,
Probab.Theor.Related Fields 184 (2022) 209-273

Published: Oct 1, 2022

e-Print:

1903.11015 [math.PR]

DOI:

10.1007/s00440-022-01142-z

View in:

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Abstract: (Springer)

The free multiplicative Brownian motion

b_{t}

is the large-N limit of the Brownian motion on

\mathsf {GL}(N;\mathbb {C}),

in the sense of

*

-distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of

b_{t}

. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region

\Sigma _{t}

that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density

W_{t}

on

\overline{\Sigma }_{t},

which is strictly positive and real analytic on

\Sigma _{t}

. This density has a simple form in polar coordinates:

\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}

where

w_{t}

is an analytic function determined by the geometry of the region

\Sigma _{t}

. We show also that the spectral measure of free unitary Brownian motion

u_{t}

is a “shadow” of the Brown measure of

b_{t}

, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

Note:

Added references to subsequent works building on these results. Made a notational change, replacing the regularization parameter x with epsilon

60B20
46L54
35F21
measure: spectral
differential equations: stochastic
Brownian motion
density
unitarity
geometry
Wigner

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