Abstract:
A bijective mapping of the space of operator-valued functions to the set of complex-valued finite additive cylindrical measures on the trajectory space is constructed and studied. Conditions are established under which the Cauchy problem for a first-order equation with a variable operator generates a two-parameter evolution family of operators. A representation of the solution to the Cauchy problem with a variable perturbed generator is obtained using the path integral of a perturbation-defined functional on the trajectory space with respect to a cylindrical pseudomeasure defined by the unperturbed two-parameter evolution family of operators.
Key words:evolution family of operators, one-parameter semigroup, finitely additive measure, Markov process, Chernoff theorem, Feynman–Kac formula.
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