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

TL;DR

This work characterizes the parabolic Martin boundary for a space-time Brownian motion with drift killed at the boundary of a moving cone, by deriving precise Green's-function asymptotics in all directions and relating harmonic functions to conditioned persistence probabilities via Doob's $h$-transform. The authors combine Malyshev's steepest-descent analysis on a kernel-induced Riemann surface with a recursive compensation approach to construct all positive harmonic functions, culminating in a complete description of the minimal Martin boundary as a parabola-arc and explicit persistence/exiting laws. They further invert Laplace transforms to obtain the exit densities and an explicit transition kernel for the killed process, providing a unified continuous-domain treatment with connections to prior discrete-work on the Martin boundary in cones. The methods yield exact formulas for edge-exit probabilities, persistence probabilities, exit-time laws, and the transition density, enhancing both the theoretical understanding and potential applications to stochastic processes with moving boundaries.

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.

Figures (4)

• Figure 1: Space time Brownian motion in $C$ and degenerate Brownian motion in $\mathbb{R}_+^2$
• Figure 2: The parabola $\mathcal{P}:=\{(p,q)\in\mathbb{R}_+^2 : K(p,q)=0 \}$ in blue and the arc of parabola $\overline{\mathcal{A}}:=\{(p(\alpha),q(\alpha)): \alpha\in[0,\pi/2] \} \subset \mathcal{P}$ which parametrizes the Martin's boundary in red
• Figure 3: Integration contour $\mathcal{C}_R$ and $Q^+(p)$ the only pole of the integrand inside $\mathcal{C}_R$
• Figure 4: Sequel of the points $(p_n,q_n)$ on the parabola $\mathcal{P}:=\{(p,q)\in\mathbb{R}^2 : K(p,q)=0\}$. On the left $(p_0,q_0)\in \mathcal{A}:=\{(p(\alpha),q(\alpha)): \alpha\in(0,\pi/2) \}$, on the right, $(p_0,q_0)=(p(0),q(0)) \in \partial \mathcal{A}$.

Theorems & Definitions (36)

• Proposition 1: Functional equation
• proof
• Proposition 2: Exit densities seen as derivative of the Green's function
• proof
• Remark 3: Harmonic functions
• Theorem 4: Asymptotics of Green's function and exit densities
• Remark 5: $h^0$ and $h^{\pi/2}$ as derivatives
• Lemma 6: Reduction to simple integrals
• proof
• Lemma 7: Asymptotics of $I_1$