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Pré-Publication, Document De Travail Année : 2024

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

Résumé

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.
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hal-04511091 , version 1 (19-03-2024)

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Paternité - Pas d'utilisation commerciale - Pas de modification

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  • HAL Id : hal-04511091 , version 1

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Jules Flin, Sandro Franceschi. Reflected brownian motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties. 2024. ⟨hal-04511091⟩
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