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Transactions of the American Mathematical Society

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



$C^{2}$-preserving strongly continuous Markovian semigroups

Author: W. M. Priestley
Journal: Trans. Amer. Math. Soc. 180 (1973), 359-365
MSC: Primary 47D05; Secondary 58G99, 60J35
MathSciNet review: 0324481
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Abstract: Let $X$ be a compact ${C^2}$-manifold. Let $||\;||,||\;||’$ 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 $||\;||$. Theorem. Assume that $\{ {P^t}\}$ preserves ${C^2}$-functions and that the restriction of $\{ {P^t}\}$ to ${C^2}(X),||\;||’$ is strongly continuous. Then ${C^2}(X) \subset \mathcal {D}$ and $A$ is a bounded operator from ${C^2}(X),||\;||’$ to $C(X),||\;||$. 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.

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Keywords: <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="${P^t}$">-function, <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${P^t}$">-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