Tohoku Mathematical Journal

On the Feller property of Dirichlet forms generated by pseudo differential operators

René L. Schilling and Toshihiro Uemura

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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.

Article information

Tohoku Math. J. (2), Volume 59, Number 3 (2007), 401-422.

First available in Project Euclid: 11 October 2007

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Zentralblatt MATH identifier

Primary: 31C25: Dirichlet spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J75: Jump processes 60G52: Stable processes 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]

Dirichlet form Beurling-Deny formula pseudo differential operator integro-differential operator Feller process stable-like process Lévy system


Schilling, René L.; Uemura, Toshihiro. On the Feller property of Dirichlet forms generated by pseudo differential operators. Tohoku Math. J. (2) 59 (2007), no. 3, 401--422. doi:10.2748/tmj/1192117985.

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