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 \(\varepsilon\) and \(\delta\). We assume that the classical parameter \(\alpha=\frac{\delta+\varepsilon - \pi}{\beta}\) 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 \(\alpha\in\mathbb{N}^*\) 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.