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

Husseini, Ryad; Kassmann, Moritz

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1099-1115.

Soit $\nu ={\left\{\nu (x,·)\right\}}_{x\in {ℝ}^d}$ 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 $\nu$. À la différence des contributions précédentes cette méthode couvre des cas où les bornes inférieures de la probabilité d'atteindre de petits ensembles dégénèrent.

Given a family of Lévy measures $\nu ={\left\{\nu (x,·)\right\}}_{x\in {ℝ}^d}$, 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 $\nu$. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.

Classification : 60J75, 35B45, 31C05, 47D07
Mots clés : jump processes, Lévy measure, Feller property, martingale problem, integro-differential operators, harmonic functions, a priori estimates

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