Skip to main content
eScholarship
Open Access Publications from the University of California

UC Berkeley

UC Berkeley Electronic Theses and Dissertations bannerUC Berkeley

Noncommutative generalized Brownian motions with multiple processes

Abstract

This thesis compiles several results under the general theme of noncommutative generalized Brownian motion with multiple processes.

In our introduction, Chapter 1, we motivate our results with some background material. Specifically, we provide brief introductions to classical Brownian motion and noncommutative probability theory, and then we connect these two subjects with a discussion of noncommutative Brownian motions.

In Chapter 2, we expand upon the framework for generalized Brownian motions with multiple processes established by Guta. In particular, we discuss symmetric Fock spaces, colored pair partitions, and colored broken pair partitions. We then prove multidimensional generalizations of some results which were proven by Guta and Maassen in the case of a single process.

In Chapter 3, we review Thoma’s Theorem on characters of the infinite symmetric group and Vershik and Kerov’s factor representations of symmetric groups. We then recall Bozejko and Guta's work on generalized Brownian motions with one process associated to the infinite symmetric group, focusing on their formula for the joint moments associated to these generalized Brownian motions. We then proceed to consider generalized Brownian motions indexed by a two-element set arise from tensor products of factor rep- resentations of the infinite symmetric group, providing a simple formula for the joint moments in this context.

In Chapter 4, we consider generalized Brownian motions connected to spherical representations of a pair of infinite symmetric groups. By defining a directed graph associated to a two-colored pair partition, we state and prove a formula for the joint moments connected with these generalized Brownian motion.

In Chapter 5, we continue the study of the generalized Brownian motions related to spherical functions of infinite symmetric groups, specializing to a one-parameter family of the spherical representations. In this setting, we show that the generalized Brownian motions give bounded operators, and we investigate related von Neumann algebras.

Finally, in Chapter 6, we generalize Guta’s q-product and prove a central limit theorem in the spirit of that of Guta.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View