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Positive solutions to discrete harmonic functions in unbounded cylinders

Fengwen Han|Lidan Wang
대한수학회 (The Korean Mathematical Society)|2024.03|대한수학회지 (Journal of the KMS)|제61권 제2호|pp. 377-393|DOI 10.4134/JKMS.j230270

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키워드

Random walkHarnack inequalitymaximum principlediscrete harmonic function
주제분류수학

초록

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.

Abstract

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.

참고문헌 (29)

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