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

The classical Liouville property says that all bounded harmonic functions in ${\mathbb{R}}^n$, i.e.\ all bounded functions satisfying $\mathrm{\Delta }f=0$, 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 Lévy processes. For generators of Lévy 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 Lévy processes.