A Dirichlet Process Characterization of RBM in a Wedge

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process Z whose state space in R is given in polar coordinates by S = {(r, θ) : r ≥ 0, 0 ≤ θ ≤ ξ} for some 0 < ξ < 2π. Let α = (θ1+θ2)/ξ, where −π/2 < θ1, θ2 < π/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 < α < 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 > α, the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p ≤ α, 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.