Colloquium: Large-N limits of N x N matrix stochastic process

Abstract: Free probability is a noncommutative analog of
probability theory that is tremendously useful for describing the
large-N limits of N x N random matrices.  Combining free probability
with ideas from stochastic analysis produces a discipline called
noncommutative stochastic analysis, which provides a framework for
describing the large-N limits of time-dependent N x N random matrices,
i.e., N x N matrix stochastic processes.  A foundational example is
Philippe Biane's introduction of free Brownian motion, the large-N
limit of Brownian motion on the space of N x N Hermitian matrices, in
the mid-1990s.  This talk will center on the more general topic of
using noncommutative stochastic analysis to describe the large-N
limits of solutions to a large class of N x N matrix stochastic
differential equations.  One goal of the talk will be to illustrate a
common theme in this research area:  the need to build spaces of
"functions of noncommuting variables" well suited to the problem under
consideration.  Though it may be useful to look up the definition of
Brownian motion on Euclidean space beforehand, you will not need to be
familiar with stochastic analysis or free probability to follow along.
This talk will be based primarily on joint work in progress with
Guillaume Cébron and Nicolas Gilliers but may also touch on both past
and ongoing joint work with various subsets of Benoît Collins, David
Jekel, Marius Junge, Todd Kemp, Félix Parraud, and Roland Speicher.

Faculty Host: John McCarthy