Skip to main content
SpringerLink
Log in

Conservativeness of Semigroups Generated by Pseudo Differntial Operators

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

Assume that the pseudo differential operator -q(x,D) generates a Fellerian or sub-Markovian semigroup. Under some natural additional conditions on the symbol -q(x,ξ) we prove that the operator -q(x,D) is conservative if and only if q(x,0)≡0.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berg, C. and Forst, G.: Potential Theory on Locally Compact Abelian Groups, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, II. Ser. 87, Berlin, 1975.

    Google Scholar 

  2. Butzer P.L. and Berens, H.: Semi-Groups of Operators and Approximation, Springer, Grundlehren Math. Wiss. Bd. 145, Berlin, 1967.

    Google Scholar 

  3. Courrège, Ph.: ‘Générateur infinitésimal d'un semi-groupe de convolution sur R n, et formule de Lévy-Khinchine’, Bull. Sci. Math. 2e sér. 88 (1964), 3–30.

    Google Scholar 

  4. Courrège, Ph.: ‘Sur la forme intégro-différentielle des opérateurs de C ∞K dans C satisfaisant au principe du maximum’, Sém. Théorie du Potentiel (1965/66), 38pp.

  5. Ethier, St.E. and Kurtz, Th.G.: Markov Processes: Characterization and Convergence, Wiley, Series in Prob. and Math. Stat., New York, 1986.

    Google Scholar 

  6. Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Studies in Math. 19, Berlin, 1994.

  7. Hoh, W.: ‘The martingale problem for a class of pseudo differential operators’, Math. Ann. 300 (1994), 121–147.

    Google Scholar 

  8. Hoh, W.: ‘Pseudo differential operators with negative definite symbols and the martingale problem’, Stoch. and Stoch. Rep. 55 (1995), 225–252.

    Google Scholar 

  9. Hoh, W. and Jacob, N.: ‘Upper bounds and conservativeness for semigroups associated with a class of Dirichlet forms generated by pseudo differential operators’, Forum Math. 8 (1996), 107–120.

    Google Scholar 

  10. Jacob, N.: ‘A class of Feller semigroups generated by pseudo differential operators’, Math. Z. 215 (1994), 151–166.

    Google Scholar 

  11. Jacob, N.: ‘Characteristic functions and symbols in the theory of Feller processes’, Warwick Preprints 46 (1995); Potential Analysis 8 (1998), 1–19.

  12. Ma, Z.M., Overbeck, L., and Röckner, M.: ‘Markov processes associated with semi-Dirichlet forms’, Osaka J. Math. 32 (1995), 97–119.

    Google Scholar 

  13. Oshima, Y.: ‘On conservativeness and recurrence criteria for Markov processes’, Potential Analysis 1 (1992), 115–131.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen, Bismarckstraβe 1 1/2, D-91054, Erlangen, Germany

    René L. Schilling

Authors
  1. René L. Schilling
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schilling, R.L. Conservativeness of Semigroups Generated by Pseudo Differntial Operators. Potential Analysis 9, 91–104 (1998). https://doi.org/10.1023/A:1008664419747

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008664419747

Search

Navigation