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.
Similar content being viewed by others
References
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.
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.
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.
Dieudonné J., Foundations of Modern Analysis, Academic Press, International Edition, New York 1969 (enlarged and corrected printing).
Ethier, St. E. and Th. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, Series in Probab. and Math. Stat., New York 1986.
Hoh, W., On perturbations of pseudo differential operators with negative definite symbol, preprint(1997).
Jacob, N., Characteristic functions and symbols in the theory of Feller processes, Potential Analysis 8(1998), 61-68.
Jacob, N., Pseudo-differential operators and Markov processes, Akademie Verlag, Mathematical Research vol. 94, Berlin 1996.
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.).
Schilling, R. L., Conservativeness of semigroups generated by pseudo differential operators, to appear in Potential Analysis.
Schilling, R. L., Growth and Hölder conditions for the sample paths of Feller processes, to appear in Probab. Theory Relat. Fields.
Author information
Authors and Affiliations
Rights 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
Issue Date:
DOI: https://doi.org/10.1023/A:1009748105208