The Liouville theorem for a class of Fourier multipliers and its connection to coupling

TL;DR

This paper characterizes when solutions to certain Fourier multiplier equations are constant or polynomial, extending classical harmonic function results to non-local operators related to Lévy processes, and applies this to coupling of space-time Lévy processes.

Contribution

It provides necessary and sufficient conditions for Liouville-type theorems for Fourier multipliers, including Lévy process generators, and connects these results to coupling methods.

Findings

Conditions for solutions to be constant or polynomial are established.

Necessary and sufficient criteria for strong Liouville theorems for Lévy generators are derived.

A coupling result for space-time Lévy processes is proved.

Abstract

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $\Delta f = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of L\'evy processes. For generators of L\'evy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time L\'evy processes.