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

Abstract

AbstractThe classical Liouville property says that all bounded harmonic functions in , that is, all bounded functions satisfying , are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator , such that the solutions to are Lebesgue a.e. constant (if is bounded) or coincide Lebesgue a.e. with a polynomial (if 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 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.