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

S. Franceschi ()
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S. Franceschi: Sorbonne Université

Journal of Theoretical Probability, 2021, vol. 34, issue 4, 1775-1810

Abstract: 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.

Keywords: Green’s function; Oblique Neumann boundary condition; Obliquely reflected Brownian motion in a wedge; Semi-martingale reflected Brownian motion; Laplace transform; Conformal mapping; Carleman boundary value problem; 30E25; 60J45; 60J65; 60H30 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01043-8

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