Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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POSITIVE SOLUTIONS TO DISCRETE HARMONIC FUNCTIONS IN UNBOUNDED CYLINDERS

  • Fengwen Han (School of Mathematics and Statistics Henan University) ;
  • Lidan Wang (School of Mathematical Sciences Jiangsu University)
  • Received : 2023.05.24
  • Accepted : 2024.01.04
  • Published : 2024.03.01
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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 ℤd. 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

Acknowledgement

The authors would like to thank the anonymous reviewer's careful reading and helpful suggestions to improve the writing of this paper. The authors would like to thank Bobo Hua for helpful discussions and suggestions.

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