Martin boundary of a reflected random walk on a half-space

Abstract

The complete representation of the Martin compactification for reflected random walks on a half-space ${\mathbb{Z}}^d\times \mathbb{N}$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in ${\mathbb{Z}}^d$ : convergence of a sequence of points $z_n\in {\mathbb{Z}}^{d-1}\times \mathbb{N}$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.

42 pages, preprint, CNRS UMR 8088

References in corpus (2)

• Bridges and networks: Exact asymptotics
• Large deviations for processes with discontinuous statistics

Cited by in corpus (4)

• Martin boundary of a killed random walk on a quadrant
• Random walks in the quarter plane, discrete harmonic functions and conformal mappings
• The t-Martin boundary of reflected random walks on a half-space
• Martin boundary of a space-time Brownian motion with drift killed at the boundary of a moving cone