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Abstract <p>We develop a method for finding the generalized expectations of unbounded random operators that are generators of strongly continuous one-parameter semigroups in a Hilbert space, based on semigroup averaging and first described in [Yu.N. Orlov, V.Zh. Sakbaev, and O.G. Smolyanov, ‘‘Unbounded random operators and Feynman formulae,’’ Izvestiya: Mathematics <b>80</b> (6), 1131–1158 (2016)]. We show that under sufficient conditions, the generalized expectation can also be found by averaging the corresponding resolvents, with the result coinciding with semigroup averaging. We also provide relatively simple examples where the expectation can be obtained by averaging resolvents but not semigroups, and vice versa.</p>

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Random One-Parameter Semigroups: On the Equivalence of Semigroup and Resolvent Averaging

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • R. Sh. Kalmetev

摘要

Abstract

We develop a method for finding the generalized expectations of unbounded random operators that are generators of strongly continuous one-parameter semigroups in a Hilbert space, based on semigroup averaging and first described in [Yu.N. Orlov, V.Zh. Sakbaev, and O.G. Smolyanov, ‘‘Unbounded random operators and Feynman formulae,’’ Izvestiya: Mathematics 80 (6), 1131–1158 (2016)]. We show that under sufficient conditions, the generalized expectation can also be found by averaging the corresponding resolvents, with the result coinciding with semigroup averaging. We also provide relatively simple examples where the expectation can be obtained by averaging resolvents but not semigroups, and vice versa.