C 2 - Preserving Strongly Continuous Markovian Semigroups

Let A' be a compact C -manifold. Let || ||, || || denote the supremum norm and the C -norm, respectively, and let \P \ be a Markovian semigroup on C(X). The semigroup's infinitesimal generator A, with domain 3), is defined by Af= lim Q t~ (P'f f), whenever the limit exists in || ||. Theorem. Assume that \P s preserves C -junctions and that the restriction of {PA to C2(X), I I' is strongly continuous. Then C (X) c 3) azzrf A is a bounded operator from C (X), || || to C(X), \\ \\. From the conclusion is obtained a representation of Af • (x) as an integrodifferential operator on C (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 A" is a Lie group and P commutes with translations. Actually, a stronger result is proved having the above theorem among its corollaries.