Liouville theorems for non-local operators

The paper characterizes some classes of pseudo-differential operators for which there are (or
there are not) non-constant bounded harmonic functions. Non-local perturbations of
Ornstein–Uhlenbeck operators and operators with dissipative coefficients are considered.
The methods used are probabilistic and based on the concept of absorption function and on a
new extension of the Bismut–Elworthy–Li formula. The probabilistic interpretation of the
Liouville theorem by means of absorption functions for general Markov processes is given as
well.