Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1900–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

Transition Probabilities for Symmetric Jump Processes


Authors: Richard F. Bass and David A. Levin
Journal: Trans. Amer. Math. Soc. 354 (2002), 2933-2953
MSC (2000): Primary 60J05; Secondary 60J35
DOI: https://doi.org/10.1090/S0002-9947-02-02998-7
Published electronically: March 11, 2002
MathSciNet review: 1895210
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider symmetric Markov chains on the integer lattice in $d$ dimensions, where $\alpha \in (0,2)$ and the conductance between $x$ and $y$ is comparable to $\vert x-y\vert^{-(d+\alpha )}$. We establish upper and lower bounds for the transition probabilities that are sharp up to constants.

References [Enhancements On Off] (What's this?)

  • [BBG] M.T. Barlow, R.F. Bass, and C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math. 53 (2000), 1007-1038. MR 2001m:35095
  • [BL] R.F. Bass and D.A. Levin, Harnack inequalities for jump processes, Potential Anal., to appear.
  • [CKS] E.A. Carlen, S. Kusuoka, and D.W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245-287. MR 88i:35066
  • [HS-C] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673-709. MR 94m:60144
  • [Kl] V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. 80 (2000), 725-768. MR 2001f:60016
  • [Km] T. Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math. 32 (1995), 833-860. MR 97d:31010
  • [Le] T. Leviatan, Perturbations of Markov processes, J. Functional Analysis 10 (1972), 309-325. MR 53:4243
  • [SZ] D.W. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 619-649. MR 98k:60125
  • [Wo] W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Univ. Press (2000). MR 2001k:60006

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60J05, 60J35

Retrieve articles in all journals with MSC (2000): 60J05, 60J35

Additional Information

Richard F. Bass
Affiliation: Department of Mathematics, The University of Connecticut, Storrs, Connecticut 06269
Email: bass@math.uconn.edu

David A. Levin
Affiliation: Department of Mathematics, The University of Connecticut, Storrs, Connecticut 06269
Address at time of publication: P.O. Box 368, Annapolis Junction, Maryland 20701-0368
Email: levin@member.ams.org

DOI: https://doi.org/10.1090/S0002-9947-02-02998-7
Keywords: Harnack inequality, jump processes, stable processes, Markov chains, transition probabilities
Received by editor(s): June 18, 2001
Received by editor(s) in revised form: December 27, 2001
Published electronically: March 11, 2002
Additional Notes: Research of the first author was partially supported by NSF grant DMS-9988496
Article copyright: © Copyright 2002 American Mathematical Society