Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties

We study a Brownian motion with drift in a wedge of angle β which is obliquely reflected on each edge along angles ε and δ. We assume that the classical parameter (Formula presented) is greater than 1 and we focus on transient cases where the process can either be absorbed at the vertex or escape to infinity. We show that (Formula presented) is a necessary and sufficient condition for the absorption probability, seen as a function of the starting point, to be written as a finite sum of terms of exponential product form. In such cases, we give expressions for the absorption probability and its Laplace transform. When (Formula presented) we find an explicit differentially-algebraic expression for the Laplace transform. Our results rely on Tutte’s invariant method and a recursive compensation approach.