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.
Liouville theorems for non-local operators
E. Priola;
2004-01-01
Abstract
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.File in questo prodotto:
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