On a Family of Generalized Wiener Spaces and Applications

Authors

Ian Pierce, University of Nebraska-LincolnFollow

Date of this Version

Spring 5-2011

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor David L. Skoug. Lincoln, Nebraska: May, 2011

Copyright (c) 2011 Ian D. Pierce

Abstract

We investigate the structure and properties of a variety of generalized Wiener spaces. Our main focus is on Wiener-type measures on spaces of continuous functions; our generalizations include an extension to multiple parameters, and a method of adjusting the distribution and covariance structure of the measure on the underlying function space.

In the second chapter, we consider single-parameter function spaces and extend a fundamental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized Wiener space of several parameters; this will be the setting for the fourth chapter, where we extend some interesting results of Cameron and Storvick. In the final chapter, we apply the work of the preceding chapters to the question of reflection principles for single-parameter and multiple-parameter Gaussian stochastic processes.