Brownian particles interacting with a Newtonian Barrier: Skorohod maps and their use in solving a PDE with free boundary, strong approximation, and hydrodynamic limits.

Abstract

In this thesis, we pioneer the use of Skorohod maps in establishing the hydrodynamic behavior of an interacting particle system. This technique has the benefit of using stochastic methods to show both existence and uniqueness of the resulting PDE with free boundary condition. In 2001, Frank Knight constructed a stochastic process modeling the one dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton's laws of motion, and the other particle being Brownian. In the first chapter we construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. The resulting PDE is similar to the solution of the Stefan problem. As mentioned, both existence and uniqueness of the PDE are done using stochastic methods; the uniqueness is done using a novel, and new, coupling method. In the second chapter, we give a strong approximation of Brownian motion with inert drift. We also determine the distribution of the maximum of the Newtonian particle via its Laplace transform. In the third chapter, we consider a random walker on the nonnegative lattice, moving in continuous time, whose transition rate is a linear function of the time the walker spends at the origin. In this way the walker is a jump process with a stochastic and adapted intensity. When Brownian scaling is introduced, such a process converges to Brownian motion with inert drift. This solves a conjecture of Burdzy and White in 2008. This convergence result is used to show two Brownian motions separated by an inert particle has a product stationary distribution on its state space where the velocity of the inert particle is Gaussian. This process of two Brownian motions separated by an inert particle was studied by White, in 2007, where the demonstration of existence for the process contains a nontrivial gap that we complete.