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 ${(x_1,x_2):0<x_1\leq\delta_0,\psi_1(x_1)<x_2<\psi_ 2(x_1)}$,with$\psi_1(0)=\psi_2(0)=0$,$\psi_1’(0)=\psi_2’(0)=0$. Given a vector field $\gamma$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin ${\gamma }^i(0):={\lim }_{x_1\to 0^{+}}\gamma (x_1,{\psi }_i(x_1))$, $i=1,2,$ and assuming there exists a vector $e^{\ast }$ 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.