Dimension-independent Harnack inequalities for subordinated semigroups

TL;DR

This paper establishes dimension-independent Harnack inequalities for subordinate semigroups, including diffusion processes under Bakry-Emery conditions, with new results for various power parameters and infinite-dimensional cases.

Contribution

It introduces novel dimension-free Harnack inequalities for subordinate semigroups, extending previous results to a broader class of diffusions and infinite-dimensional examples.

Findings

Harnack inequalities hold for subordinate semigroups with specific power parameters.

Dimension-free inequalities are established for diffusions satisfying Bakry-Emery conditions.

Infinite-dimensional examples demonstrate the applicability of the inequalities.

Abstract

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $\alpha$ satisfies a dimension-free Harnack inequality provided $\alpha \in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $\alpha \in (0,1).$ Some infinite-dimensional examples are also presented.