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

## TL;DR

This paper derives an explicit integral expression for the moment generating function of Green's functions for obliquely reflected Brownian motion with drift in the first quadrant, using a novel kernel functional equation and boundary value problem approach.

## Contribution

It introduces a new kernel functional equation linking Green's functions inside the quadrant and on its edges, enabling explicit integral formulas.

## Key findings

- Explicit integral expression for Green's functions' moment generating function
- Development of a new kernel functional equation for the process
- Solution of a boundary value problem using conformal mapping

## 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.