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December 2021 Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary
Philip A. Ernst, Sandro Franceschi
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Ann. Appl. Probab. 31(6): 2991-3016 (December 2021). DOI: 10.1214/21-AAP1681

Abstract

Let π be the occupancy density of an obliquely reflected Brownian motion in the half plane and let (ρ,α) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of π(ρ,α) as ρ with α(0,π). We find explicit functions a, b, c such that

π(ρ,α)ρa(α)ρb(α)ec(α)ρ.

This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy measure and we obtain an explicit integral expression for π. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.

Funding Statement

The first-named author thanks Rice University’s Dobelman Family Junior Chair; he also gratefully acknowledges the support of ARO-YIP-71636-MA, NSF Grant DMS-1811936, and ONR Grant N00014-18-1-2192.

Acknowledgments

We are deeply grateful to Professor J. Michael Harrison for sharing this problem with us as well as for providing some initial ideas about it. We are also grateful to Irina Kourkova, Masakiyo Miyazawa, and Kilian Raschel for helpful discussions about this problem. We acknowledge, with thanks, Dongzhou Huang for helpful feedback.

Citation

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Philip A. Ernst. Sandro Franceschi. "Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary." Ann. Appl. Probab. 31 (6) 2991 - 3016, December 2021. https://doi.org/10.1214/21-AAP1681

Information

Received: 1 April 2020; Revised: 1 December 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-AAP1681

Subjects:
Primary: 60J60 , 60K25
Secondary: 30D05 , 90B22

Keywords: exact asymptotics , Green’s function , Laplace transform , Martin boundary , obliquely reflected Brownian motion in a half-plane , Occupancy density , saddle-point method , stationary distribution

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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