Abstract
Difference estimates and Harnack inequalities for mean zero, finite variance random walks with infinite range are considered. An example is given to show that such estimates and inequalities do not hold for all mean zero, finite variance random walks. Conditions are then given under which such results can be proved.
Similar content being viewed by others
References
Bass, R. F. and Khoshnevisan, D. (1992). Local times on curves and uniform invariance principles.Prob. Th. Rel. Fields 92, 465–492.
Burdzy, K. and Lawler, G. (1990). Non-intersection exponents for Brownian paths. Part I. Existence and Invariance Principle,Prob. Th. Rel. Fields 84, 393–410.
Griffin, P. S. and McConnell, T. R. (1992). On the position of a random walk at the time of first exit from a sphere.Ann. Prob. 20, 825–854.
Lawler, G. (1989). Low-density expansion for a two-state random walk in a random environment.J. Math. Phys. 30, 145–157.
Lawler, G. (1991).Intersections of Random Walks, Birkhäuser-Boston.
Lawler, G. (1991). Estimates for Differences and Harnack Inequality for Difference Operators Coming from Random Walks with Symmetric, Spatially Inhomogeneous Increments,Proc. London Math. Soc. 63(3), 552–568.
Ney, P. and Spitzer, F. (1966). The Martin Boundary for Random Walk,Trans. Amer. Math. Soc. 121, 116–132.
Polaski, T. (1991). Estimates for Differences and Harnack's Inequality for Functions Harmonic with Respect to Random Walks, Ph.D. Dissertation, Duke University.
Spitzer, F. (1976)Principles of Random Walk, Springer-Verlag.
Author information
Authors and Affiliations
Additional information
Research supported by the National Science Foundation.
Rights and permissions
About this article
Cite this article
Lawler, G.F., Polaski, T.W. Harnack inequalities and difference estimates for random walks with infinite range. J Theor Probab 6, 781–802 (1993). https://doi.org/10.1007/BF01049175
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01049175