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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.12661 |
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| _version_ | 1866929517831913472 |
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| author | Flin, Jules Franceschi, Sandro |
| author_facet | Flin, Jules Franceschi, Sandro |
| contents | We study a Brownian motion with drift in a wedge of angle $β$ which is obliquely reflected on each edge along angles $\varepsilon$ and $δ$. We assume that the classical parameter $α=\frac{δ+\varepsilon - π}β$ 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 $α\in\mathbb{N}^*$ 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 $α\in\mathbb{Z}+\fracπβ\mathbb{Z}$ we find explicit D-algebraic expression for the Laplace transform. Our results rely on Tutte's invariant method and on a recursive compensation approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties Flin, Jules Franceschi, Sandro Probability We study a Brownian motion with drift in a wedge of angle $β$ which is obliquely reflected on each edge along angles $\varepsilon$ and $δ$. We assume that the classical parameter $α=\frac{δ+\varepsilon - π}β$ 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 $α\in\mathbb{N}^*$ 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 $α\in\mathbb{Z}+\fracπβ\mathbb{Z}$ we find explicit D-algebraic expression for the Laplace transform. Our results rely on Tutte's invariant method and on a recursive compensation approach. |
| title | Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties |
| topic | Probability |
| url | https://arxiv.org/abs/2403.12661 |