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 wh...
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description | 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. |
doi_str_mv | 10.48550/arxiv.2403.12661 |
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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. 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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.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.2403.12661</doi><oa>free_for_read</oa></addata></record> |
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subjects | Absorption Angle of reflection Brownian motion Laplace transforms Mathematics - Probability |
title | Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties |
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