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

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.