Feynman–Kac Formulas for Difference-differential Equations of Retarded Type

We study the bijection of the space of one-parameter families of bounded linear operators in a space of functions on a coordinate space onto the set of complex-valued finitely additive cylindrical measures in the space of trajectories in the coordinate space. From the space of measures on the space of trajectories, a set of measures representing semigroups is distinguished. A procedure is defined for constructing Feynman measures on the space of trajectories representing semigroups generated by a linear functional differential equation with delay. Using the Feynman–Kac formulas and a measure on the space of trajectories corresponding to an unperturbed semigroup, the perturbation of a semigroup by a bounded potential on the space of trajectory values is studied. A perturbed semigroup is defined by integrating the perturbation functional on the trajectory space over the Feynman cylindrical measure corresponding to the unperturbed semigroup.