Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant

TL;DR

This paper analyzes the behavior of obliquely reflected Brownian motion in a quadrant, deriving PDEs, boundary conditions, and asymptotics for escape and absorption probabilities, including conditions for product-form solutions.

Contribution

It introduces a new geometric criterion called the dual skew symmetry condition that characterizes when absorption probabilities are exponential and have a product form.

Findings

Derived PDEs and boundary conditions for escape probability.

Identified a geometric condition for product-form and exponential absorption probabilities.

Provided explicit integral expressions and asymptotics for escape probabilities.

Abstract

We consider an obliquely reflected Brownian motion $Z$ with positive drift in a quadrant stopped at time $T$, where $T:=\inf \{ t>0 : Z(t)=(0,0) \}$ is the first hitting time of the origin. Such a process can be defined even in the non-standard case where the reflection matrix is not completely-$\mathcal{S}$. We show that in this case the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in a finite time. Given an arbitrary starting point $(u,v)$ in the quadrant, we consider the escape (resp. absorption) probabilities $\mathbb{P}_{(u,v)}[T=\infty]$ (resp. $\mathbb{P}_{(u,v)}[T<\infty]$). We establish the partial differential equations and the oblique Neumann boundary conditions which characterize the escape probability and provide a functional equation satisfied by the Laplace transform of the escape probability. We then give asymptotics for the absorption probability in the simpler case where the starting point in the quadrant is $(u,0)$. We exhibit a remarkable geometric condition on the parameters which characterizes the case where the absorption probability has a product form and is exponential. We call this new criterion the dual skew symmetry condition due to its natural connection with the skew symmetry condition for the stationary distribution. We then obtain an explicit integral expression for the Laplace transform of the escape probability. We conclude by presenting exact asymptotics for the escape probability at the origin.