Journal of the
Korean Mathematical Society JKMS

In this paper, we are mainly concerned with two-sided estimates for transition probabilities of symmetric Markov chains on ${ \mathbb{Z}  }^d$, whose one-step transition probability is comparable to $|x-y|^{-d}\phi_j(|x-y|)^{-1}$ with $\phi_j$ being a positive regularly varying function on $[1,\infty)$ with index $\alpha\in [2,\infty)$. 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).

Keywords: Symmetric Markov chain, transition probability, L\'evy measure, Dirichlet form, Davies method

MSC numbers: Primary 60J05, 60J35

Supported by: The research is supported by the National Natural Science Foundation of China (Nos. 11831014 and 12071076) and the Education and Research Support Program for Fujian Provincial Agencies.