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 $\beta$ which is obliquely reflected on each edge along angles $\epsilon$ and $\delta$. We assume that the classical parameter $\alpha =\frac{\delta +\epsilon -\pi }{\beta }$ 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 $\alpha \in {\mathbb{N}}^{\ast }$ 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 $\alpha \in \mathbb{Z}+\frac{\pi }{\beta }\mathbb{Z}$ we find explicit D-algebraic expression for the Laplace transform. Our results rely on Tutte’s invariant method and on a recursive compensation approach.