On Particle Interaction Models

Abstract

This dissertation deals with three problems in Stochastic Analysis which broadly involve interactions, either between particles (Chapters 1 and 2), or between particles and the boundary of a C2 domain (Chapter 3). In Chapter 1, we introduce a new model called the Brownian Conga Line. It is a random curve evolving in time, generated when a particle performing a two dimensional Gaussian random walk leads a long chain of particles connected to each other by cohesive forces. We approximate the discrete Conga line in some sense by a smooth random curve and subsequently study the properties of this smooth curve. In Chapter 2 (joint work with Chris Hoffman), we investigate a Random Mass Split- ting Model and the closely related random walk in a random environment (RWRE) whose heat kernel at time t turns out to be the mass splitting distribution at t. We prove a quenched invariance principle (QIP) and consequently a quenched central limit theorem for this RWRE using techniques from Rassoul-Agha and Sepp al ainen [12] which in turn was based on the work of Kipnis and Varadhan [7] and others. In Chapter 3, we deal with a particle performing a Brownian motion inside a bounded C2 domain with reflection and diffusion at the boundary. We call this model Brow- nian Motion with Boundary diffusion following [1], and study its properties.