When is a Markovian Semigroup Induced by a Semiflow?

Abstract

, for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf2 = 2fZf. If either (1) there exists δ > 0 such that (Tt f)2∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g2∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φt from X into X — such that Ttf(x) = f(φt (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf2 = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φt are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f2∈ D and Zf2 = 2fZf} = {f ∈ D| Tt(f2)-(Ttf)2 is o(t2) in C(X)}. In the general case N is an algebra closed under composition with any C1-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C2-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ½ φ″[f], Provided Z is a local operator and D is an algebra. In this case the form itself enjoys the relation = φ′ [f] ψ′[g] , for C2functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let Tt = etz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||Tt - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a2) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The etz , -∞ < t < ∞, is a group of Markovian operators.