Consider the deterministic Skorokhod equation in the closed first quadrant: \(\begin{aligned} X_t=x_0+ f(t)+\int _0^t\textbf{v}(X_s)\, dL_s,\end{aligned}\) where \(f:[0,\infty )\rightarrow \mathbb {R}^2\) is a continuous function, \(f(0)=0\) , \(X_t\) 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 \(\textbf{v}\) equals \((-a_1,1)\) on the positive x axis, equals \((1,-a_2)\) on the positive y axis, and \(\textbf{v}(0)\) points into the closed first quadrant. Let \(\theta _i=\arctan a_i\) , \(i=1,2\) . Suppose that \(\theta _1+\theta _2<\pi /2\) , \(\theta _2<0\) , \(\theta _1>-\theta _2>0\) , \(|a_1a_2|>1\) and \(\begin{aligned}\frac{\log |a_1|+\log |a_2|}{a_1+a_2}>\pi /2.\end{aligned}\) We prove that for almost every trajectory of standard 2-dimensional Brownian motion \(B_t\) , the Skorokhod equation with \(f(t)\equiv B_t\) has at least two solutions.
