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A dual skew symmetry for transient reflected Brownian motion in an orthant

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Abstract

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison (Adv Appl Probab 10:886–905, 1978), which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.

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Acknowledgements

We are grateful to Philip Ernst and to John Michael Harrison for very interesting discussions about topics related to this article. We thank an anonymous referee for very useful remarks and suggestions.

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Authors and Affiliations

  1. Laboratoire SAMOVAR, Télécom SudParis, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Sandro Franceschi

  2. Laboratoire Angevin de Recherche en Mathématiques, CNRS, Université d’Angers, 49000, Angers, France

    Kilian Raschel

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  1. Sandro Franceschi
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Correspondence to Kilian Raschel.

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This project has received funding from the ANR RESYST and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 759702.

We are honored to dedicate this paper to Professor Masakiyo Miyazawa and to address him our warmest wishes for his 75th birthday. His beautiful contributions in the field of reflected Brownian motion (in particular [12]) are at the origin of our interest to this area and remain a great source of inspiration.

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Franceschi, S., Raschel, K. A dual skew symmetry for transient reflected Brownian motion in an orthant. Queueing Syst 102, 123–141 (2022). https://doi.org/10.1007/s11134-022-09853-9

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