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 $R^n$, that is, 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 $m\left(D\right)$, such that the solutions $f$ to $m\left(D\right)f=0$ are Lebesgue a.e. constant (if $f$ is bounded) or coincide Lebesgue a.e. with a polynomial (if $f$ is polynomially bounded). 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.