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.
The research of V. Zh. Sakbaev (Sections 2 and 3) was supported by the Russian Science Foundation under grant 19-11-00320 and performed at Steklov Mathematical Institute of Russian Academy of Sciences. The research of Yu. N. Orlov and O. G. Smolyanov (Sections 4–6) was performed within the joint project with the Laboratory of Infinite-Dimensional Analysis and Mathematical Physics at the Faculty of Mechanics and Mathematics, Moscow State University, and supported by the Russian Academic Excellence Project “5-100.”
Citation:
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Mathematical physics and applications, Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov, Trudy Mat. Inst. Steklova, 306, Steklov Math. Inst. RAS, Moscow, 2019, 210–226; Proc. Steklov Inst. Math., 306 (2019), 196–211
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\paper Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups
\inbook Mathematical physics and applications
\bookinfo Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 306
\pages 210--226
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
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This publication is cited in the following 13 articles:
V. M. Busovikov, Yu. N. Orlov, V. Zh. Sakbaev, “Unitary representation of walks along random vector fields and the Kolmogorov–Fokker–Planck equation in a Hilbert space”, Theoret. and Math. Phys., 218:2 (2024), 205–221
S. V. Dzhenzher, V. Zh. Sakbaev, “Quantum Law of Large Numbers for Banach Spaces”, Lobachevskii J Math, 45:6 (2024), 2485
Oleg E. Galkin, Ivan D. Remizov, “Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators”, Isr. J. Math., 2024
R. Sh. Kalmetev, “Usrednenie po Chernovu lineinykh differentsialnykh uravnenii”, Preprinty IPM im. M. V. Keldysha, 2023, 010, 12 pp.
J. E. Gough, Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Markov approximations of the evolution of quantum systems”, Dokl. Math., 105:2 (2022), 92–96
K. Yu. Zamana, V. Zh. Sakbaev, “Kompozitsii nezavisimykh sluchainykh operatorov i svyazannye s nimi differentsialnye uravneniya”, Preprinty IPM im. M. V. Keldysha, 2022, 049, 23 pp.
V. Zh. Sakbaev, E. V. Shmidt, V. Shmidt, “Limit distribution for compositions of random operators”, Lobachevskii J. Math., 43:7 (2022), 1740
R. Sh. Kalmetiev, Yu. N. Orlov, V. Zh. Sakbaev, “Chernoff iterations as an averaging method for random affine transformations”, Comput. Math. Math. Phys., 62:6 (2022), 996–1006
K. Yu. Zamana, “Averaging of random orthogonal transformations of domain of functions”, Ufa Math. J., 13:4 (2021), 23–40
J. E. Gough, Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Random quantization of Hamiltonian systems”, Dokl. Math., 103:3 (2021), 122–126
Yu. N. Orlov, V. Zh. Sakbaev, E. V. Shmidt, “Operator approach to weak convergence of measures and limit theorems for random operators”, Lobachevskii J. Math., 42:10, SI (2021), 2413–2426
Yu. N. Orlov, V. Zh. Sakbaev, D. V. Zavadskii, “Operator Random Walks and Quantum Oscillator”, Lobachevskii J. Math., 41:4, SI (2020), 676–685
K. Yu. Zamana, V. Zh. Sakbaev, O. G. Smolyanov, “Stochastic processes on the group of orthogonal matrices and evolution equations describing them”, Comput. Math. Math. Phys., 60:10 (2020), 1686–1700