Existence and uniqueness of reflecting diffusions in cusps

TL;DR

This paper establishes the existence and uniqueness of reflecting diffusions in a 2D domain with a cusp at the origin, addressing challenges posed by the domain's geometry and boundary reflection directions.

Contribution

It proves weak and strong existence and uniqueness of solutions for reflecting diffusions in cusped domains, introducing new scaling and coupling techniques.

Findings

Weak existence and uniqueness at the cusp

Strong existence and uniqueness away from the cusp

New methods for analyzing reflecting diffusions in complex domains

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 $\gamma$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $\gamma^i(0):=\lim_{x_1\rightarrow 0^{+}}\gamma (x_1,\psi_i(x_1))$, $ 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.