A Dirichlet Process Characterization of RBM in a Wedge

TL;DR

This paper characterizes reflected Brownian motion in a wedge as a Dirichlet process, providing a decomposition into Brownian motion and a zero-energy process, and analyzes its path variation properties.

Contribution

It proves RBM in a wedge is a Dirichlet process with a specific decomposition and explores its path variation and Skorokhod problem solutions.

Findings

RBM in a wedge is a Dirichlet process for 1<alpha<2.

The process Y has finite p-variation for p>alpha and infinite for p<=alpha.

On excursions, (Z,Y) satisfies the standard Skorokhod problem, but not on the entire horizon.

Abstract

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process Z whose state space in R^2 is given in polar coordinates by S={(r,theta): r >= 0, 0 <= theta <= xi} for some 0 < xi < 2 pi. Let alpha= (theta_1+theta_2)/xi, where -pi/2 < theta_1,theta_2 < pi/2 are the directions of reflection of Z off each of the two edges of the wedge as measured from the corresponding inward facing normal. We prove that in the case of 1 < alpha < 2, RBM in a wedge is a Dirichlet process. Specifically, its unique Doob-Meyer type decomposition is given by Z=X+Y, where X is a two-dimensional Brownian motion and Y is a continuous process of zero energy. Furthermore, we show that for p > alpha , the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p <= alpha, the strong p-variation of Y is infinite on [0,T] whenever Z has been started from the origin. We also show that on excursion intervals of Z away from the origin, (Z,Y) satisfies the standard Skorokhod problem for X. However, on the entire time horizon (Z,Y) does not satisfy the standard Skorokhod problem for X, but nevertheless we show that it satisfies the extended Skorkohod problem.