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

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  • S. Franceschi

    (Sorbonne Université)

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.

Suggested Citation

  • S. Franceschi, 2021. "Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1775-1810, December.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01043-8
    DOI: 10.1007/s10959-020-01043-8
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    References listed on IDEAS

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