Journal of the
Korean Mathematical Society JKMS

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