Harnack inequalities in infinite dimensions

TL;DR

This paper investigates the validity of Harnack inequalities for harmonic functions in infinite-dimensional spaces, showing their failure for some operators and their validity for others, with implications for stochastic processes.

Contribution

It demonstrates the non-existence of Harnack inequalities for the infinite dimensional Laplacian and certain Ornstein-Uhlenbeck processes, but establishes their validity for operators in Hörmander's form.

Findings

Harnack inequality does not hold for the infinite dimensional Laplacian.

Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes.

Harnack inequality holds for the infinite dimensional operators in Hörmander's form.

Abstract

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in H\"{o}rmander's form. In this case a Harnack inequality does hold.