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.

中文翻译：

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象限中具有斜诺依曼边界条件的格林函数

我们研究了瞬态情况下平面第一象限中带有漂移的半鞅斜反射布朗运动。我们的主要结果确定了该过程格林函数的矩生成函数的一般显式积分表达式。为此，我们建立了一个新的核函数方程，将象限内及其边缘的格林函数的矩生成函数连接起来。这让人想起经常出现的情况，其中函数方程源自表征平稳分布的基本伴随关系。这个方程将我们引向非齐次卡尔曼边值问题。它的分辨率为基于轮廓积分和保角映射的矩生成函数提供了一个公式。