On harmonic functions of symmetric Lévy processes

Mimica, Ante

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 214-235.

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.

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.

Classification : 60J45, 60J75, 60J25
Mots clés : geometric stable process, Green function, harmonic function, Lévy process, modulus of continuity, subordinator, subordinate brownian motion

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