Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ C\sp{2}$-preserving strongly continuous Markovian semigroups


Author: W. M. Priestley
Journal: Trans. Amer. Math. Soc. 180 (1973), 359-365
MSC: Primary 47D05; Secondary 58G99, 60J35
DOI: https://doi.org/10.1090/S0002-9947-1973-0324481-5
MathSciNet review: 0324481
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a compact $ {C^2}$-manifold. Let $ \vert\vert\;\vert\vert,\vert\vert\;\vert\vert'$ denote the supremum norm and the $ {C^2}$-norm, respectively, and let $ \{ {P^t}\} $ be a Markovian semigroup on $ C(X)$. The semigroup's infinitesimal generator $ A$, with domain $ \mathcal{D}$, is defined by $ Af = {\lim _{t \to 0}}{t^{ - 1}}({P^t}f - f)$, whenever the limit exists in $ \vert\vert\;\vert\vert$.

Theorem. Assume that $ \{ {P^t}\} $ preserves $ {C^2}$-functions and that the restriction of $ \{ {P^t}\} $ to $ {C^2}(X),\vert\vert\;\vert\vert'$ is strongly continuous. Then $ {C^2}(X) \subset \mathcal{D}$ and $ A$ is a bounded operator from $ {C^2}(X),\vert\vert\;\vert\vert'$ to $ C(X),\vert\vert\;\vert\vert$.

From the conclusion is obtained a representation of $ Af \cdot (x)$ as an integrodifferential operator on $ {C^2}(X)$. The representation reduces to that obtained by Hunt [Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293] in case $ X$ is a Lie group and $ {P^t}$ commutes with translations.

Actually, a stronger result is proved having the above theorem among its corollaries.


References [Enhancements On Off] (What's this?)

  • [1] E. B. Dynkin, Markov processes. Vol. 1, Fizmatgiz, Moscow, 1963; English transl., Die Grundlehren der math. Wissenschaften, Band 122, Academic Press, New York; Springer-Verlag, Berlin, 1965. MR 33 #1886; #1887. MR 0193670 (33:1886)
  • [2] E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [3] G. A. Hunt, Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293. MR 18, 54. MR 0079232 (18:54a)
  • [4] E. Nelson, Dynamical theories of Brownian motion, Princeton Univ. Press, Princeton, N. J., 1967. MR 35 #5001. MR 0214150 (35:5001)
  • [5] -, Representation of a Markovian semigroup and its infinitesimal generator, J. Math. Mech. 7 (1958), 977-987. MR 20 #7224. MR 0100796 (20:7224)
  • [6] W. M. Priestley, Markovian semigroups on non-commutative $ {P^t}$-algebras: An elementary study, Ph.D. Thesis, Princeton University, University Microfilms, Ann Arbor, Mich., 1972.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D05, 58G99, 60J35

Retrieve articles in all journals with MSC: 47D05, 58G99, 60J35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0324481-5
Keywords: $ {P^t}$-function, $ {P^t}$-manifold, Markovian operator, strong derivative, Banach-Steinhaus theorem, Markovian semigroup, infinitesimal generator, integro-differential operator, normal derivative of harmonic extension
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society