Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary

Abstract

Let $\mathit{\pi }$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let $(\mathit{\rho },\mathit{\alpha })$ be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of $\mathit{\pi }(\mathit{\rho },\mathit{\alpha })$ as $\mathit{\rho }\to \infty$ with $\mathit{\alpha }\in (0,\mathit{\pi })$. We find explicit functions a, b, c such that

$$\mathit{\pi }(\mathit{\rho },\mathit{\alpha })\underset{\mathit{\rho }\to \infty }{\sim }\mathit{a}(\mathit{\alpha }){\mathit{\rho }}^{\mathit{b}(\mathit{\alpha })}{\mathit{e}}^{-\mathit{c}(\mathit{\alpha })\mathit{\rho }}.$$

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 $\mathit{\pi }$. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.