The authors study semidynamical systems and develop elements of a (probabilistic) potential theory for semidynamical systems, the associated deterministic semigroup and resolvent. In particular, the notions of hitting and terminal times and multiplicative functionals are treated (pretty much in the vein of R. M. Blumenthal and R. K. Getoor’s [“Markov processes and potential theory” (Series of Monographs and Textbooks 29. New York-London) (1968; Zbl 0169.49204)].
The main result of the paper is a characterization of multiplicative semigroups over a Lusin space: if the excessive functions are a min-stable and separate points and if the \(0\)-resolvent of the constant function {1} satisfies some additional isomorphism property, then the multiplicative semigroup is deterministic, i.e., it arises as canonical deterministic semigroup of some semidynamical system. The proof of this result is based on a theorem on semigroups \((Q_t)_t\) subordinate to a canonical deterministic semigroup \((H_t)_t\) of a semidynamical system: if \(Q_tf \leq H_tf\) for all positive measurable \(f\) (i.e., subordinate), then \(Q_t = M_tH_t\) for some multiplicative functional \(M_t\). If \(Q_t\) is also deterministic, then \(M_t\) is given by \(1_{[0,T)}\), where \(T\) is a terminal time.