A note on the Green's function for the transient random walk without killing on the half lattice, orthant and strip

TL;DR

This paper derives an exact formula for the Green's function of a transient random walk on various subspaces of the lattice, extending understanding of random walk behavior without boundary killing.

Contribution

It provides a new explicit formula relating Green's functions on subspaces to those on the full lattice, for dimensions d ≥ 3.

Findings

Explicit Green's function formulas for half lattice, orthant, and strip.

Connections established between subspace and full lattice Green's functions.

Enhanced understanding of random walk behavior without boundary killing.

Abstract

In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $\mathbb{Z}^d$, $d\ge 3$.