Skip to main content
Log in

Conservativeness and Extensions of Feller Semigroups

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

Let \(\{ T_t \} _{t \geqslant 0} \) denote a Feller semigroup on \(C_\infty (\mathbb{R}^n )\), and \(\{ \tilde T_t \} _{t \geqslant 0} \) itsextension to the bounded measurable functions. We show that\(T_t 1 \in C_b (\mathbb{R}^n )\). If the generator of the semigroup is a pseudo-differential operator we can restate this condition in terms of the symbol. As a by-product, we obtain necessary and sufficient conditions for the conservativeness of the semigroup which are again expressed through the symbol.

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

Access this article

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 G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, II. Ser. Bd. 87, Berlin 1975.

    Google Scholar 

  2. Bony, J.-M., Courrège, Ph. et P. Priouret, Semi-groupes de Feller sur une variété à bord compacte et problème aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier, Grenoble 18.2 (1968), 369-521.

    Google Scholar 

  3. Courrège, Ph., Sur la forme intégro-différentielle des opérateurs de C \(_K^\infty \) dans C satisfaisant au principe du maximum, Sém. Théorie du Potentiel (1965/66) 38 p.

  4. Dieudonné J., Foundations of Modern Analysis, Academic Press, International Edition, New York 1969 (enlarged and corrected printing).

    Google Scholar 

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

    Google Scholar 

  6. Hoh, W., On perturbations of pseudo differential operators with negative definite symbol, preprint(1997).

  7. Jacob, N., Characteristic functions and symbols in the theory of Feller processes, Potential Analysis 8(1998), 61-68.

    Google Scholar 

  8. Jacob, N., Pseudo-differential operators and Markov processes, Akademie Verlag, Mathematical Research vol. 94, Berlin 1996.

    Google Scholar 

  9. Rogers, L. C. G. and D. Williams, Diffusions, Markov Processes, and Martingales. Volume one: Foundations, Wiley, Series in Probab. Math. Stat., Chichester 1994 (2nd ed.).

    Google Scholar 

  10. Schilling, R. L., Conservativeness of semigroups generated by pseudo differential operators, to appear in Potential Analysis.

  11. Schilling, R. L., Growth and Hölder conditions for the sample paths of Feller processes, to appear in Probab. Theory Relat. Fields.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schilling, R.L. Conservativeness and Extensions of Feller Semigroups. Positivity 2, 239–256 (1998). https://doi.org/10.1023/A:1009748105208

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009748105208

Navigation