Abstract
In this paper, we consider a large class of subordinate random walks X on the integer lattice \(\mathbb {Z}^d\) via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for nonnegative harmonic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards, Gaithersburg (1972)
Bass, R.F., Kumagai, T.: Symmetric Markov Chains on \(\mathbb{Z}^d\) with Unbounded Range. Transactions of the American Mathematical Society 360, 2041–2075 (2008)
Bass, R.F., Levin, D.A.: Transition Probabilities for Symmetric Jump Processes. Transactions of the American Mathematical Society 354, 2933–2953 (2002)
Bendikov, A., Cygan, W.: \(\alpha \) - Stable Random Walk has Massive Thorns, arXiv:1307.4947v3 (2016)
Bendikov, A., Cygan, W., Trojan, B.: Limit Theorems for Random Walks, arXiv:1504.01759v2 (2015)
Bendikov, A., Saloff-Coste, L.: Random walks on groups and discrete subordination. Mathematische Nachrichten 285, 580–605 (2012)
Hebisch, W., Saloff-Coste, L.: Gaussian Estimates for Markov Chains and Random Walks on Groups. Annals of Probability 21, 673–709 (1993)
Kim, P., Song, R., Vondraček, Z.: Global uniform boundary Harnack principle with explicit decay rate and its application. Stochastic Processes and their Applications 124, 235–267 (2014)
Lawler, G.F.: Intersections of Random Walks. Birkhäuser, Basel (1996)
Lawler, G.F., Polaski, W.: Harnack inequalities and Difference Estimates for Random Walks with Infinite Range. Journal of Theoretical Probability 6, 781–802 (1993)
Mimica, A., Kim, P.: Harnack inequalities for subordinate Brownian motions. Electronic Journal of Probability 37, 1–23 (2012)
Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)
Schilling, R., Song, R., Vondraček, Z.: Bernstein functions: Theory and applications, 2nd edn. De Gruyter, Berlin (2012)
Acknowledgements
This work has been supported in part by Croatian Science Foundation under the project 3526.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mimica, A., Šebek, S. Harnack Inequality for Subordinate Random Walks. J Theor Probab 32, 737–764 (2019). https://doi.org/10.1007/s10959-018-0821-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-018-0821-5