Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary. (English) Zbl 1482.60109

Ernst, Philip A.; Franceschi, Sandro

Ann. Appl. Probab. 31, No. 6, 2991-3016 (2021).

Summary: Let \(\boldsymbol{\pi}\) be the occupancy density of an obliquely reflected Brownian motion in the half plane and let \((\rho ,\alpha)\) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of \(\boldsymbol{\pi}(\rho, \alpha)\) as \(\rho \to \infty\) with \(\alpha \in (0,\pi)\). We find explicit functions \(a, b, c\) such that \[ \boldsymbol{\pi} (\rho, \alpha) \underset{\rho \to \infty}{\sim} a(\alpha) \rho^{b(\alpha)} e^{-c(\alpha) \rho}. \] 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 \(\boldsymbol{\pi}\). We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.

MSC:

60J65	Brownian motion
60E07	Infinitely divisible distributions; stable distributions
60K25	Queueing theory (aspects of probability theory)
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
90B22	Queues and service in operations research

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

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