Feynman–Kac formulas for solutions of evolution equations. Part I. Generalized random processes and operator families

A generalized random process with values in a measurable space is defined as a complex-valued finite additive cylindrical measure on the space of trajectories with values in the measurable space. Using this extension of the concept of a random process, we aim to obtain a representation of solutions to the evolutionary equation by averaging functionals on the space of trajectories of a random process. For this purpose, a bijective mapping of the space of operator valued functions into a set of complex valued finite additive cylindrical measures on the trajectory space is constructed and investigated. Limit theorems for generalized random processes are obtained. In the second part of the survey, the application of the constructed bijective mapping to the obtaining of perturbed semigroups and evolutionary families of operators in the form of Feynman–Kac formulas will be considered.