Operator valued measures and perturbations of semi-groups (Q788140)

The paper provides a refined mathematical treatment of those problems of mathematical physics which may be interpreted as a perturbation of one given semi-group of operators by another one, such as a diffusion process perturbed by the creation and/or destruction of the diffusing substances, for example. For that purpose the well known Feynman-Kac formula which expresses the total amount of the substance at any time after the beginning of the process as the integral of a function depending on the reaction rate with respect to the Wiener measure [see \textit{M. Kac}, Probability and related topics in physical sciences (1959; Zbl 0087.330)] is generalized by replacing the Wiener measure by a certain measure- valued measure, naturally related to the initial distribution via the diffusion semi-group. Therefore the generalized formula gives the actual distribution of the substance and not merely the total amount of it. The solution uses integration of scalar-valued functions with respect to vector-valued measures, an account of which is given in section 1 of the paper. The main tool is a Fubini-Tonelli type theorem on product integration with respect to vector measure and real-valued measure. The classical Kac-formula follows from the generalized one with new proof. Applications are given to diffusion and dynamical systems.