Jump processes, ℒ-harmonic functions, continuity estimates and the Feller property

Jump processes, ℒ-harmonic functions, continuity estimates and the Feller property

Ryad Husseini and Moritz Kassmann

Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 4 (2009), 1099-1115.

Abstract

Given a family of Lévy measures ν={ν(x, ⋅)}_x∈ℝ^d, the present work deals with the regularity of harmonic functions and the Feller property of corresponding jump processes. The main aim is to establish continuity estimates for harmonic functions under weak assumptions on the family ν. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.

Résumé

Soit ν={ν(x, ⋅)}_x∈ℝ^d une famille de mesures de Lévy, ce travail étudie la régularité de fonctions harmoniques et la propriété de Feller du processus de saut correspondant. Le but principal est d’établir des estimations de continuité pour les fonctions harmoniques sous des conditions faibles sur la famille ν. À la différence des contributions précédentes cette méthode couvre des cas où les bornes inférieures de la probabilité d’atteindre de petits ensembles dégénèrent.

First Page: Show

Primary Subjects: 60J75, 35B45, 31C05, 47D07

Keywords: Jump processes; Lévy measure; Feller property; Martingale problem; Integro-differential operators; Harmonic functions; A priori estimates

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Permanent link to this document: http://projecteuclid.org/euclid.aihp/1257529894
Digital Object Identifier: doi:10.1214/08-AIHP208
Zentralblatt MATH identifier: 05758872
Mathematical Reviews number (MathSciNet): MR2572166

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