Abstract
We show that the subordination induced by a convolution semigroup (subordination in the sense of Bochner) of a C 0-semigroup of sub-Markovian operators on an L p space is actually associated to the subordination of a right (Markov) process. As a consequence, we solve the martingale problem associate with the L p-infinitesimal generator of the subordinate semigroup. We also prove quasi continuity properties for the elements of the domain of the L p-generator of the subordinate semigroup. It turns out that an enlargement of the base space is necessary. A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process. We use several analytic and probabilistic potential theoretical tools.
Similar content being viewed by others
References
Albeverio, S., Rüdiger, B.: Infinite-dimensional stochastic differential equations obtained by subordination and related Dirichlet forms. J. Funct. Anal., 204, 122–156 (2003)
Beznea, L., Boboc, N.: Potential Theory and Right Processes, Springer Series, Mathematics and Its Applications (572), Kluwer, Dordrecht, 2004
Beznea, L., Boboc, N.: On the tightness of capacities associated with sub-Markovian resolvents. Bull. London Math. Soc., 37, 899–907 (2005)
Beznea, L., Boboc, N., Röckner, M.: Markov processes associated with L p-resolvents and applications to stochastic differential equations on Hilbert space. J. Evol. Eq., 6, 745–772 (2006)
Beznea, L., Boboc, N., Röckner, M.: Quasi-regular Dirichlet forms and L p-resolvents on measurable spaces. Potential Anal., 25, 269–282 (2006)
Beznea, L., Röckner, M.: From resolvents to càdlàg processes through compact excessive functions and applications to singular SDE on Hilbert spaces. Bull. Sci. Math., 135, 844–870 (2011)
Beznea, L., Röckner, M.: Applications of compact superharmonic functions: path regularity and tightness of capacities. Compl. Anal. Oper. Theory, 5, 731–741 (2011)
Bliedtner, J., Hansen, W.: Potential Theory — An Analytic and Probabilistic Approach to Balayage, Springer Verlag, Berlin, 1986
Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory, Academic Press, New York, 1968
Bouleau, N.: Quelques résultats probabilistes sur la subordination au sens de Bochner. In: Lect. Notes Math., 1061, Springer, 1984, 54–81
Dellacherie, C., Meyer, P.-A.: Probabilités et potentiel. Ch. I–IV, IX–XI, XII–XVI, Hermann, Paris, 1975, 1983, 1987
Gordina, M., Röckner, M., Wang, F.-Y.: Dimension-independent Harnack inequalities for subordinated semigroups. Potential Anal., 34, 293–307 (2011)
Hirsch, F.: Générateurs étendus et subordination au sens de Bochner. In: Lect. Notes Math., 1061, Springer, 1984, 134–156
Hmissi, F., Hmissi, M.: On subordination of resolvents and application to right processes. Stochastics, 81, 345–353 (2009)
Jacob, N.: Pseudo Differential Operators & Markov Processes (Vols. I, II, III), Imperial College Press, 2005
Jacob, N., Schilling, R. L.: Subordination in the sense of S. Bochner — an approach through pseudo differential operators. Math. Nachr., 178, 199–231 (1996)
Jacob, N., Schilling, R. L.: Some Dirichlet spaces obtained by subordinate reflected diffusions. Rev. Mat. Iberoamer., 15, 59–91 (1999)
Ma, Z. M., Röckner, M.: Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Springer-Verlag, New York, 1992
Nelson, E.: A functional calculus using singular Laplace integrals. Trans. Amer. Math. Soc., 88, 400–413 (1958)
Schilling, R. L., Song, R., Vondraček, Z.: Bernstein Functions: Theory and Applications, Walter de Gruyter, 2010
Sharpe, M.: General Theory of Markov Processes, Academic Press, Boston, 1988
Steffens, J.: Excessive measures and the existence of right semi-groups and processes. Trans. Amer. Math. Soc., 311, 267–290 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by a grant of the Romanian National Authority for Scientific Research, CNCS — UEFISCDI (Grant No. PN-II-ID-PCE-2011-3-0045)
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Lupaşcu, O. Subordination in the sense of Bochner of L p-semigroups and associated Markov processes. Acta. Math. Sin.-English Ser. 30, 187–196 (2014). https://doi.org/10.1007/s10114-014-2751-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-014-2751-1