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Liouville theorems for non-local operators. (English) Zbl 1063.31003
From the authors’ summary: 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.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60J75 Jump processes (MSC2010)
47D07 Markov semigroups and applications to diffusion processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
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