Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TWO-SIDED ESTIMATES FOR TRANSITION PROBABILITIES OF SYMMETRIC MARKOV CHAINS ON ℤd

  • Zhi-He Chen (School of Mathematics and Statistics Fujian Normal University)
  • Received : 2022.03.06
  • Accepted : 2022.12.12
  • Published : 2023.05.01
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Abstract

In this paper, we are mainly concerned with two-sided estimates for transition probabilities of symmetric Markov chains on ℤd, whose one-step transition probability is comparable to |x - y|-dϕj (|x - y|)-1 with ϕj being a positive regularly varying function on [1, ∞) with index α ∈ [2, ∞). For upper bounds, we directly apply the comparison idea and the Davies method, which considerably improves the existing arguments in the literature; while for lower bounds the relation with the corresponding continuous time symmetric Markov chains are fully used. In particular, our results answer one open question mentioned in the paper by Murugan and Saloff-Coste (2015).

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Acknowledgement

I would like to express my great gratitude to the anonymous referee for his/her corrections and insightful comments, which improve considerably my work.

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