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

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 at the origin. Such a process can be defined even in the non-standard case in which the reflection matrix is not completely-S. We show in this case that the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in finite time. Given an arbitrary starting point (u,v) in the quadrant, we consider the escape (resp. absorption) probabilities P_(u,v)[T=∞] (resp. P_(u,v)[T<∞]). 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. Asymptotics for the absorption probability in the simpler case in which the starting point in the quadrant is (u,0) are then given. We proceed to show a geometric criterion on the parameters which characterizes the case in which 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 and conclude by presenting exact asymptotics for the escape probability at the origin.