Feynman–Kac formulas for solutions of evolution equations. Part II. Evolution families and functional integrals

An analog of Feynman–Kac formula for a perturbation of a C0-semigroup generator by a bounded multiplicative operator is obtained by using the bijection (which had been constructed in Part I [1]) of a space of operator valued function to a space of cylindrical measures on the space of trajectories. Classes of integrable with respect to a cylindrical measure functions on the space of trajectories are defined. Conditions of correct solvability of a Cauchy problem for a first order evolution equation with time dependent generator are obtained. The representation of a solution of this Cauchy problem is given in the form of functional integral with respect to a nostationar Markovian cylindrical measure on the space of trajectories.