Feynman calculus for random operator-valued functions and their applications

The Feynman–Chernoff iteration of a random semigroup of bounded linear operators in the Hilbert space has been considered. The convergence of mean values of the Feynman–Chernoff iteration of a random semigroup has been studied. The estimates of the deviation of compositions of the independent identically distributed random semigroup from its mean value have been obtained as the large numbers law for the sequence of compositions of the independent random semigroup has been investigated. The relationship between the semigroup properties of the mean values of the random operator-valued function and the property of independence of the increments of the random operator-valued function has been analyzed. The property of asymptotic independence of the increments of the Feynman–Chernoff iteration of the random semigroup has been discussed. The independization of the random operator-valued function has been defined as the map of this random operator function into the sequence of random operator-valued functions, which has asymptotically independent increments. The examples of independization (which is similar to the Feynman–Chernoff iteration) of the random operator-valued function have been given.