Existence and uniqueness of reflecting diffusions in cusps

Abstract

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 $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $ i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, 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.