# Transition probability estimates for subordinate random walks

## TL;DR

This paper derives global estimates for the transition probabilities of subordinate random walks on integer lattices, using parabolic Harnack inequalities and bounds on the continuous-time kernel, under specific conditions on the Bernstein function.

## Contribution

It provides new global transition probability estimates for subordinate random walks based on Bernstein functions, extending previous results to a broader class of subordinations.

## Key findings

- Established global transition probability estimates for subordinate random walks.
- Applied parabolic Harnack inequality and kernel bounds in the analysis.
- Extended understanding of transition behaviors under specific Bernstein function conditions.

## Abstract

Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $\phi$ we consider a random walk $S^\phi_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $\phi$ at zero we establish global estimates for the transition probabilities of the random walk $S^\phi_n$. 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.