The Liouville theorem and linear operators satisfying the maximum principle

On the Liouville property for non-local Lévy generators

“…holds; •, • denotes the (real) dual pairing used in the theory of distributions. An excellent account on the history and the importance of the Liouville property can be found in the paper [1] by Alibaud et al…”

“…Like [1] we are interested in the analogue of (1) for a class of non-local operators with constant 'coefficients'. Recall that 1 2 ∆ is the infinitesimal generator of Brownian motion.…”

“…Like [1] we are interested in the analogue of (1) for a class of non-local operators with constant 'coefficients'. Recall that 1 2 ∆ is the infinitesimal generator of Brownian motion. This is a diffusion process with stationary and independent increments and continuous paths.…”

“…The paper [1] by Alibaud et al contains another characterization of the Liouville property for Lévy generators using completely different methods: It is based on the characteristic triplet (b, Q, ν) appearing in the Lévy-Khintchine representation (3) of ψ. If we combine our Theorem with the result of [1], we arrive at an interesting description of the zero-set of a negative definite function.…”

“…We need the following notation from [1]. Let Σ be the positive semidefinite square root of Q and denote by σ 1 , .…”