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Asymptotics for the Green’s functions of a transient reflected Brownian motion in a wedge

Author

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  • Sandro Franceschi

    (Institut Polytechnique de Paris, Télécom SudParis, Laboratoire SAMOVAR)

  • Irina Kourkova

    (Sorbonne Universite, Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001)

  • Maxence Petit

    (Sorbonne Universite, Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001)

Abstract

We consider a transient Brownian motion reflected obliquely in a two-dimensional wedge. A precise asymptotic expansion of Green’s functions is found in all directions. To this end, we first determine a kernel functional equation connecting the Laplace transforms of the Green’s functions. We then extend the Laplace transforms analytically and study its singularities. We obtain the asymptotics applying the saddle point method to the inverse Laplace transform on the Riemann surface generated by the kernel.

Suggested Citation

  • Sandro Franceschi & Irina Kourkova & Maxence Petit, 2024. "Asymptotics for the Green’s functions of a transient reflected Brownian motion in a wedge," Queueing Systems: Theory and Applications, Springer, vol. 108(3), pages 321-382, December.
    • Handle: RePEc:spr:queues:v:108:y:2024:i:3:d:10.1007_s11134-024-09925-y
    DOI: 10.1007/s11134-024-09925-y
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    References listed on IDEAS

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    1. Andrey Sarantsev, 2017. "Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1200-1223, September.
    2. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
    3. J. Dai & J. Harrison, 2012. "Reflecting Brownian motion in three dimensions: a new proof of sufficient conditions for positive recurrence," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(2), pages 135-147, April.
    4. Ernst, Philip A. & Franceschi, Sandro & Huang, Dongzhou, 2021. "Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 634-670.
    5. Sandro Franceschi & Kilian Raschel, 2022. "A dual skew symmetry for transient reflected Brownian motion in an orthant," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 123-141, October.
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