Existence and uniqueness of reflecting diffusions in cusps

We consider stochastic differential equations with (oblique) reflection in a $2$-dimensional domain that has a cusp at the origin, i..e. in a neighborhood of the origin has the form $\{(x_1,x_2):0origin, defining directions of reflection at the origin $\gamma^i(0):=\lim_{x_1\rightarrow 0^{+}}\gamma (x_1,\psi_i(x_1))$, $ i=1,2,$ and assuming there exists a vector $e^{*}$ such that $\langle e^{*},\gamma^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin. Our proof uses a new scaling result and a coupling argument.