# Harnack inequality for subordinate random walks

## TL;DR

This paper establishes Harnack inequalities and estimates for transition probabilities and Green functions for a broad class of subordinate random walks on integer lattices, advancing understanding of their harmonic functions.

## Contribution

It introduces new estimates and proves the Harnack inequality for subordinate random walks with Laplace exponents satisfying specific scaling conditions.

## Key findings

- Derived transition probability estimates
- Established Green function bounds
- Proved Harnack inequality for harmonic functions

## Abstract

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.