Liouville theorems for non-local operators

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. r

Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form

“…(see [11] The operator satisfies (H1) * and (H2), and it is left translation invariant on K (see [1,11] is contained in the class of degenerate Ornstein-Uhlenbeck operators studied by Priola and Zabczyk [12], where a Liouville theorem for bounded solutions is proved. Finally, it is easy to recognize that there is no Lie group structure in R 3 leaving left translation invariant the operator ᏸ.…”

Uniqueness for Integro-PDE in Hilbert Spaces

“…[9,60]. More general equations with additional first order terms were studied in [49,64]. In particular [64] contains the so called BEL formula for the gradient of the solutions.…”

“…More general equations with additional first order terms were studied in [49,64]. In particular [64] contains the so called BEL formula for the gradient of the solutions. Nonlinear parabolic problems with non-local operators, corresponding to optimal stopping problems were investigated in [75] using the theory of maximal monotone operators.…”

Estimates of the derivatives for parabolic operators with unbounded coefficients

“…If ω 1,p > 0, in general, such a result fails. Counterexamples are also given in [18] in the one-dimensional case.…”

“…As already stressed in the Introduction, such a result fails in general. We refer the reader to [18] for further details. Proof.…”