Pathwise non-uniqueness for Brownian motion in a quadrant with oblique reflection

Abstract

Consider the Skorokhod equation in the closed first quadrant: \[ X_t=x_0+ B_t+\int_0^t{\bf v}(X_s)\, dL_s,\] where $B_t$ is standard 2-dimensional Brownian motion, $X_t$ takes values in the quadrant for all $t$, and $L_t$ is a process that starts at 0, is non-decreasing and continuous, and increases only at those times when $X_t$ is on the boundary of the quadrant. Suppose ${\bf v}$ equals $(-a_1,1)$ on the positive $x$ axis, equals $(1,-a_2)$ on the positive $y$ axis, and ${\bf v}(0)$ points into the closed first quadrant. Let $\theta_i=\arctan a_i$, $i=1,2$. It is known that there exists a solution to the Skorokhod equation for all $t\geq 0$ if and only if $\theta_1+\theta_2<\pi/2$ and moreover the solution is unique if $|a_1a_2|<1$. Suppose now that $\theta_1+\theta_2<\pi/2$, $\theta_2<0$, $\theta_1>-\theta_2>0$ and $|a_1a_2|>1$. We prove that for a large class of $(a_1,a_2)$, namely those for which \[\frac{\log|a_1|+\log|a_2|}{a_1+a_2}>\pi/2,\] pathwise uniqueness for the Skorokhod equation fails to hold.