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

Sandro Franceschi

TL;DR

The article addresses the problem of characterizing Green's functions for a transient obliquely reflected Brownian motion in the quadrant by deriving an explicit integral formula for the moment generating function $\psi$ of Green's measure and its boundary components $\psi_1$, $\psi_2$. The main approach introduces a kernel functional equation $-\gamma(\theta)\psi(\theta)=\gamma_1(\theta)\psi_1(\theta_2)+\gamma_2(\theta)\psi_2(\theta_1)+e^{\theta\cdot x}$ and proceeds through kernel analysis, holomorphic continuation, and a Carleman boundary value problem, ultimately solving the problem via a conformal glueing function and a contour-integral representation. The paper provides a complete analytic framework, including a detailed BVP resolution and a discussion of decoupling functions to simplify the problem in special cases, extending potential theory for non-smooth domains with oblique reflection. These results yield a computable integral formula for Green's functions and open avenues for asymptotic analysis, invariant measures, and extensions to wedges or higher dimensions with potential applications in stochastic networks and related fields.

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.

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

TL;DR

The article addresses the problem of characterizing Green's functions for a transient obliquely reflected Brownian motion in the quadrant by deriving an explicit integral formula for the moment generating function of Green's measure and its boundary components , . The main approach introduces a kernel functional equation and proceeds through kernel analysis, holomorphic continuation, and a Carleman boundary value problem, ultimately solving the problem via a conformal glueing function and a contour-integral representation. The paper provides a complete analytic framework, including a detailed BVP resolution and a discussion of decoupling functions to simplify the problem in special cases, extending potential theory for non-smooth domains with oblique reflection. These results yield a computable integral formula for Green's functions and open avenues for asymptotic analysis, invariant measures, and extensions to wedges or higher dimensions with potential applications in stochastic networks and related fields.

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.

Paper Structure

This paper contains 30 sections, 18 theorems, 115 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results and strategy
  4. Perspectives
  5. Structure of the paper
  6. Transient SRBM in the quadrant
  7. Definition
  8. Recurrence and transience
  9. A new functional equation
  10. Functional equation
  11. Kernel
  12. Holomorphic continuation
  13. A boundary value problem
  14. Boundary and domain
  15. Carleman boundary value problem
  16. ...and 15 more sections

Key Result

Proposition 1

Let us define ${\textcolor{black}{Z=(Z(t),t\geqslant 0)}}$ a SRBM with drift in the quarter plane $\mathbb{R}_+^2$ associated to $(\Sigma, \mu, R)$ as the semi-martingale such that for $t\in \mathbb{R}_+$ we have where $x$ is the starting point, $W$ is a planar Brownian motion starting from $0$ and of covariance $\Sigma$ and for $i=1,2$ the coordinate ${\textcolor{black}{L_i(t)}}$of $L\textcolor{

Figures (11)

  • Figure 1: Reflection vectors and drift
  • Figure 2: Existence conditions
  • Figure 3: Recurrence and transience conditions according to the parameters
  • Figure 4: Ellipse $\gamma=0$, straight lines $\gamma_1=0$ and $\gamma_2=0$ and intersection points $\theta^*$ and $\theta^{**}$
  • Figure 5: Curve $\mathcal{R}$ defined in \ref{['eq:curve_definition1']} in green and domain $\mathcal{G}_\mathcal{R}$ in blue
  • ...and 6 more figures

Theorems & Definitions (39)

  • Proposition 1: Existence and uniqueness
  • Proposition 2: Transience properties
  • Proposition 3: Transience and recurrence
  • Remark 4
  • Proposition 5: Functional equation
  • proof
  • Lemma 6: Holomorphic continuation
  • proof
  • Lemma 7
  • proof
  • ...and 29 more