Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups

Abstract

We study sequences of compositions of independent identically distributed random one-parameter semigroups of linear transformations of a Hilbert space and the asymptotic properties of the distributions of such compositions when the number of terms in the composition tends to infinity. To study the expectation of such compositions, we apply the Feynman-Chernoff iterations obtained via Chernoff’s theorem. By the Feynman-Chernoff iterations we mean prelimit expressions from the Feynman formulas; the latter are representations of one-parameter semigroups or related objects in terms of the limit of integrals over Cartesian powers of an appropriate space, or some generalizations of such representations. In particular, we study the deviation of the values of compositions of independent random semigroups from their expectation and examine the validity for such compositions of analogs of the limit theorems of probability theory such as the law of large numbers. We obtain sufficient conditions under which any neighborhood of the expectation of a composition of n random semigroups contains the (random) value of this composition with probability tending to one as n → ∞ (it is this property that is viewed as the law of large numbers for compositions). We also present examples of sequences of independent random semigroups for which the law of large numbers for compositions fails.