The Brown measure of a family of free multiplicative Brownian motions

Hall, Brian C.; Ho, Ching-Wei

doi:10.1007/s00440-022-01166-5

The Brown measure of a family of free multiplicative Brownian motions

Brian C. Hall
(
- Notre Dame U.
)
,
Ching-Wei Ho
(
- Notre Dame U. and
- Academia Sinica, Taiwan
)

Apr 16, 2021

72 pages

Published in:

Probab.Theor.Related Fields 186 (2023) 3-4, 1081-1166,
Probab.Theor.Related Fields 186 (2023) 1081

Published: Aug 1, 2023

e-Print:

2104.07859 [math.PR]

DOI:

10.1007/s00440-022-01166-5

View in:

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Abstract: (Springer)

We consider a family of free multiplicative Brownian motions

b_{s,\tau }

parametrized by a real variance parameter s and a complex covariance parameter

\tau .

We compute the Brown measure

\mu _{s,\tau }

of

ub_{s,\tau },

where u is a unitary element freely independent of

b_{s,\tau }.

We find that

\mu _{s,\tau }

has a simple structure, with a density in logarithmic coordinates that is constant in the

\tau

-direction. These results generalize those of Driver–Hall–Kemp and Ho–Zhong for the case

\tau =s.

We also establish a remarkable “model deformation phenomenon,” stating that all the Brown measures with s fixed and

\tau

varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton–Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver–Hall–Kemp, our methods are substantially different at both the technical and conceptual level.

Note:

Author final version. To appear in Probability Theory and Related Fields. 72 pages with 14 figures

46L54 Free probability and free operator algebras
60B20 Random matrices (probabilistic aspects)
35F21 Hamilton-Jacobi equations
58J65 Diffusion processes and stochastic analysis on manifolds
boundary condition: unitarity
family
Brownian motion
structure
density

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