An Introduction to Brownian Motion, Wiener Measure, and Partial Differential Equations

Name:    Prof. Michael Mascagni                 
Address: Department of Computer Science and
         Department of Mathematics and
	 Department of Scientific Computing and
         Graduate Program in Molecular Biophysics	
         Florida State University
         Tallahassee, FL  32306-4530  USA
         AND
         Information Technology Laboratory
         Applied and Computational Mathematics Division
         100 Bureau Drive M/S 8910
         National Institute of Standards and Technology (NIST)
         Gaithersburg, MD  20899-8910  USA

Offices: 498 Dirac Science Library/207A Love Building (FSU) Building 225/Room B154 (NIST)
Phone:   +1.850.644.3290 (FSU) +1.301.975.2051 (NIST)
FAX:     +1.850.644.0058
e-mail:  mascagni@fsu.edu (FSU) mascagni@nist.gov (NIST)

Title: An Introduction to Brownian Motion, Wiener Measure, and Partial Differential Equations

This is a long introduction to how stochastic processes and partial differential equations are related. We begin with an introduction to Brownian motion, which is a Markoffian stochastic process. We give some of it's elementary properties and present the notion of expectation with respect to Brownian motion, and introduce Wiener measure. We then compute some Wiener integrals, and derive an explicit representation of Brownian motion. We then introduce the Feynman-Kac formulas for the probabilistic representation of partial differential equations. Time permitting, we explore some asymptotic results in action- and entropy-asymptotics. We also introduce the idea of probabilistic potential theory.