Liouville's theorems for L\'evy operators

TL;DR

This paper establishes Liouville's theorem for positive harmonic functions related to general Le9vy operators, showing they are mixtures of exponentials, and extends previous results to signed functions with growth restrictions, including a counterexample.

Contribution

It proves Liouville's theorem for positive harmonic functions of Le9vy operators and extends the theorem to signed functions under growth conditions, providing new insights and a counterexample.

Findings

Positive harmonic functions are mixtures of exponentials.

Liouville's theorem extends to signed functions with growth restrictions.

Counterexample shows theorem does not hold for all signed harmonic functions.

Abstract

Let $L$ be a L\'evy operator. A function $h$ is said to be harmonic with respect to $L$ if $L h = 0$ in an appropriate sense. We prove Liouville's theorem for positive functions harmonic with respect to a general L\'evy operator $L$: such functions are necessarily mixtures of exponentials. For signed harmonic functions we provide a fairly general result, which encompasses and extends all Liouville-type theorems previously known in this context, and which allows to trade regularity assumptions on $L$ for growth restrictions on $h$. Finally, we construct an explicit counterexample which shows that Liouville's theorem for signed functions harmonic with respect to a general L\'evy operator $L$ does not hold.