Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2024; 61(2): 377-393

Online first article February 19, 2024      Printed March 1, 2024

Copyright © The Korean Mathematical Society.

Positive solutions to discrete harmonic functions in unbounded cylinders

Fengwen Han, Lidan Wang

Henan University; Jiangsu University


In this paper, we study the positive solutions to a discrete harmonic function for a random walk satisfying finite range and ellipticity conditions, killed at the boundary of an unbounded cylinder in $\mathbb{Z}^d$. We first prove the existence and uniqueness of positive solutions, and then establish that all the positive solutions are generated by two special solutions, which are exponential growth at one end and exponential decay at the other. Our method is based on maximum principle and a Harnack type inequality.

Keywords: Random walk, discrete harmonic function, maximum principle, Harnack inequality

MSC numbers: 35J25, 35R02