Probabilistic representations of operator semigroups. A unifying approach (Q789084)

In the present paper it is shown that all known probabilistic representation theorems for strongly continuous operator semigroups can be subsumed under a single probabilistic representation formula which is based on a weak law of large numbers for a random number of summands. The general theorem gives further rise to certain product representation formulas with not necessarily equal factors which seem to be new in the literature. As a main tool, a rigorous approach to integration theory of operator semigroup-valued random variables via an extended form of Pettis' integral is used.