Transition probability estimates for subordinate random walks

Abstract

Let be a symmetric simple random walk on the integer lattice ℤ. For a Bernstein function we consider a random walk which is subordinated to. Under a certain assumption on the behaviour of at zero we establish global estimates for the transition probabilities of the random walk. The main tools that we apply are a parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.

Subordinated Markov processes: sharp estimates for heat kernels and Green functions

“…For instance, in the Euclidean space with ( ) = we get the well-known limitation < 2. In Section 6 we present possible applications of Theorems 5.2 and 5.4, which improves results obtained previously in [21,[36][37][38].…”

“…Let us notice that Theorem 6.1 applied to Z improves results obtained by Cygan and Šebek [21], not only by dropping the CBF assumption on but also by covering the cases when the first step is comparable to .…”

“…[5, 10, 14-18, 24-26, 29, 31-33]. Lastly, there are relatively few articles on heavy tailed discrete time Markov chains [21,[36][37][38].…”