Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On harmonic functions of symmetric Lévy processes

Ante Mimica

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Abstract

We consider some classes of Lévy processes for which the estimate of Krylov and Safonov (as in (Potential Anal. 17 (2002) 375–388)) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Hölder continuity estimates of harmonic functions. Despite the failure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from (Probab. Theory Related Fields 135 (2006) 547–575) and obtain asymptotic behavior of the Green function and the Lévy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.

Résumé

On considère des classes de processus de Lévy pour lesquels les estimations de Krylov et Safonov (comme dans (Potential Anal. 17 (2002) 375–388)) ne sont pas verifiées donc il n’est pas possible d’utiliser la technique standard d’itération pour obtenir a priori des estimations de continuité Hölder pour des fonctions harmoniques. Bien qu’il soit impossible d’appliquer cette méthode, on obtient des estimations a priori de régularité de fonctions harmoniques pour ces processus. De plus, on étend les résultats de (Probab. Theory Related Fields 135 (2006) 547–575) et on obtient les comportements asymptotiques de la fonction de Green et de la densité de Lévy pour une grande classe de mouvements browniens subordonnés, où l’exposant de Laplace du subordinateur correspondant est une fonction à variation lente.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 214-235.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545272

Digital Object Identifier
doi:10.1214/12-AIHP508

Mathematical Reviews number (MathSciNet)
MR3161529

Zentralblatt MATH identifier
1298.60054

Subjects
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J75: Jump processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Geometric stable process Green function Harmonic function Lévy process Modulus of continuity Subordinator Subordinate Brownian motion

Citation

Mimica, Ante. On harmonic functions of symmetric Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 214--235. doi:10.1214/12-AIHP508. https://projecteuclid.org/euclid.aihp/1388545272


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