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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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.

Article information

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

Dates
First available in Project Euclid: 11 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1192117985

Digital Object Identifier
doi:10.2748/tmj/1192117985

Mathematical Reviews number (MathSciNet)
MR2365348

Zentralblatt MATH identifier
1141.31006

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.tmj/1192117985


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References

  • R.,F. Bass and M. Kaß mann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc. 357 (2004), 837--850.
  • R.,F. Bass and D.,A. Levin, Harnack inequalities for jump processes, Potential Anal. 17 (2002), 375--388.
  • M. Fukushima, Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc. 162 (1971), 185--224.
  • M. Fukushima, N. Jacob and H. Kaneko, On $(r,2)$-capacities for a class of elliptic pseudo differential operators, Math. Ann. 293 (1992), 343--348.
  • M. Fukushima and H. Kaneko, $(r,p)$-capacities for general Markovian semigroups, Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), 41--47, Res. Notes Math. 124, Pitman, Boston, Mass., 1985.
  • M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Stud. Math. 19, Berlin 1994.
  • W. Hoh, A symbolic calculus for pseudo differential operators generating Feller semigroups, Osaka J. Math. 35 (1998), 798--820.
  • W. Hoh, Pseudo differential operators with negative definite symbols of variable order, Rev. Mat. Iberoamericana 16 (2000), 219--241.
  • Y. Isozaki and T. Uemura, A family of symmetric stable-like processes and its global path properties, Probab. Math. Statist. 24 (2004), 145--164.
  • N. Jacob, Dirichlet forms and pseudo differential operators, Expo. Math. 6 (1988), 313--351.
  • N. Jacob, Pseudo differential operators and Markov processes, Vols. 1 & 2, Imperial College Press, London, 2001-2.
  • H. Kaneko, On $(r,p)$-capacities for Markov processes, Osaka J. Math. 23 (1986), 325--336.
  • T. Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math. 25 (1988), 697--728.
  • Y. Ōshima, On conservativeness and recurrence criteria for Markov processes, Potential Anal. 1 (1992), 115--131.
  • L.,C.,G. Rogers and D. Williams, Diffusions, Markov processes and martingales, vol. 1 (2nd. ed.), Cambridge Mathematical Library, Cambridge, 2000.
  • R.,L. Schilling, Conservativeness of semigroups generated by pseudo differential operators, Potential Anal. 9 (1998), 91--104.
  • R.,L. Schilling, Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Related Fields 112 (1998), 565--611.
  • R.,L. Schilling, Dirichlet operators and the positive maximum principle, Integral Equations Operator Theory 41 (2001), 74--92.
  • R. Song and Z. Vondraček, Harnack inequalities for some classes of Markov processes, Math. Z. 246 (2004), 177--202.
  • T. Uemura, On some path properties of symmetric stable-like processes for one dimension, Potential Anal. 16 (2002), 79--91.
  • T. Uemura, On symmetric stable-like processes: some path properties and generators, J. Theoret. Probab. 17 (2004), 541--555.