Martin boundary of a space-time Brownian motion with drift killed at the boundary of a moving cone

TL;DR

This paper characterizes the Martin boundary and harmonic functions of a space-time Brownian motion with drift killed at a moving cone boundary, providing explicit formulas and asymptotic behaviors.

Contribution

It introduces a novel combination of analytical and recursive methods to analyze the Martin boundary and harmonic functions for this specific stochastic process.

Findings

Determined the parabolic Martin boundary for the process.

Derived explicit formulas for transition kernels and exit probabilities.

Analyzed asymptotics of Green's functions in all directions.

Abstract

We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t) killed at the moving boundary of the cone {(t,x):0<x<t}. This article determines the parabolic Martin boundary and all harmonic functions associated with this process. To that end, the asymptotics of Green's functions are determined along all directions. We also find the exit probabilities at the edges, the probability of remaining in the cone forever and the laws of the exit point and exit time. From this, we derive an explicit formula for the transition kernel of the process. These results arise from two different methods initially introduced to study random walks. An analytical approach, developed in the 1970s by Malyshev and based on the steepest descent method on a Riemann surface, is used to determine the asymptotics of the Green's functions. A recursive compensation approach, inspired by the method developed in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic functions.