February 2024 Reflected random walks and unstable Martin boundary
Irina Ignatiouk-Robert, Irina Kurkova, Kilian Raschel
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 549-587 (February 2024). DOI: 10.1214/22-AIHP1326


We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp. non-rational), then the Martin boundary is countable, homeomorphic to Z{±} (resp. uncountable, homeomorphic to R{±}). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.

Nous introduisons une famille de marches aléatoires en dimension deux, réfléchies au bord du quart de plan positif, et étudions leur frontière de Martin. Tandis que leur frontière minimale est systématiquement une union de deux points, nous montrons que la frontière de Martin complète est intrinsèquement instable, au sens suivant : lorsqu’un certain paramètre associé au modèle s’avère rationnel (respectivement non rationnel), la frontière de Martin est alors dénombrable et homéomorphe à Z{±} (respectivement non dénombrable et homéomorphe à R{±}). De tels phénomènes d’instabilité sont rares dans la littérature. Les démonstrations contiennent plusieurs estimées précises pour des fonctions de Green de marches aléatoires réfléchies avec probabilité de fuite le long des axes, possédant en outre un nombre infini de domaines d’inhomogénéité. Nos méthodes mélangent des techniques probabilistes avec une approche analytique pour des marches aléatoires avec grands pas en dimension deux.

Funding Statement

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 759702.


We thank Gerold Alsmeyer, Onno Boxma and Dmitry Korshunov for bibliographic suggestions. The last author would like to warmly thank Elisabetta Candellero, Steve Melczer and Wolfgang Woess for many discussions at the initial stage of the project. We thank the associate editor and the two anonymous referees for their very careful readings and their numerous suggestions.


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Irina Ignatiouk-Robert. Irina Kurkova. Kilian Raschel. "Reflected random walks and unstable Martin boundary." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 549 - 587, February 2024. https://doi.org/10.1214/22-AIHP1326


Received: 24 September 2020; Revised: 22 September 2022; Accepted: 23 September 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718390
Digital Object Identifier: 10.1214/22-AIHP1326

Primary: 31C35 , 60G50
Secondary: 31C20 , 60J45 , 60J50

Keywords: functional equation , Green function , Martin boundary , reflected random walk

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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