Tohoku Mathematical Journal
2007
September
SECOND SERIES VOL. 59, NO. 3

Tohoku Math. J.
59 (2007), 401-422

Title ON THE FELLER PROPERTY OF DIRICHLET FORMS GENERATED BY PSEUDO DIFFERENTIAL OPERATORS

Dedicated to Professor Masatoshi Fukushima on his seventieth birthday
Author René L. Schilling and Toshihiro Uemura

(Received February 28, 2006, revised August 24, 2006)
Abstract. We show that a large class of regular symmetric Dirichlet forms is generated by pseudo differential operators. We calculate the symbols which are closely related to the semimartingale characteristics (Lévy system) of the associated stochastic processes. Using the symbol we obtain estimates for the mean sojourn time of the process for balls. These estimates and a perturbation argument enable us to prove Hölder regularity of the resolvent and semigroup; this entails that the semigroup has the Feller property.

2000 Mathematics Subject Classification. Primary 31C25; Secondary 60J35, 60J75, 60G52, 47G30.
Key words and phrases. Dirichlet form, Beurling-Deny formula, pseudo differential operator, integro-differential operator, Feller process, stable-like process, Lévy system.

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