Green's Functions with Oblique Neumann Boundary Conditions in the Quadrant

Abstract

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green's functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green's functions inside the quadrant and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the basic adjoint relationship which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the moment generating function in terms of contour integrals and a conformal mapping.

Journal of Theoretical Probability, Springer, 2020

References in corpus (6)

• Atlas models of equity markets
• Intervals in the greedy Tamari posets
• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Positive recurrence of reflecting Brownian motion in three dimensions
• Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary
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Cited by in corpus (4)

• Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant
• A dual skew symmetry for transient reflected Brownian motion in an orthant
• Asymptotics for the Green's functions of a transient reflected Brownian motion in a wedge
• Martin boundary of a space-time Brownian motion with drift killed at the boundary of a moving cone