Transition probability estimates for subordinate random walks

Abstract

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

To appear in Mathematische Nachrichten