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Chernoff Iterations as an Averaging Method for Random Affine Transformations

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Abstract

For functions defined on a finite-dimensional vector space, we study compositions of their independent random affine transformations that represent a noncommutative analogue of random walks. Conditions on iterations of independent random affine transformations are established that are sufficient for convergence to a group solving the Cauchy problem for an evolution equation of shift along the averaged vector field and sufficient for convergence to a semigroup solving the Cauchy problem for the Fokker–Planck equation. Numerical estimates for the deviation of random iterations from solutions of the limit problem are presented. Initial-boundary value problems for differential equations describing the evolution of functionals of limit random processes are formulated and studied.

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Authors and Affiliations

  1. Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    R. Sh. Kalmetev, Yu. N. Orlov & V. Zh. Sakbaev

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  1. R. Sh. Kalmetev
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  2. Yu. N. Orlov
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  3. V. Zh. Sakbaev
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Correspondence to R. Sh. Kalmetev, Yu. N. Orlov or V. Zh. Sakbaev.

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The authors declare that they have no conflicts of interest.

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Translated by I. Ruzanova

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Kalmetev, R.S., Orlov, Y.N. & Sakbaev, V.Z. Chernoff Iterations as an Averaging Method for Random Affine Transformations. Comput. Math. and Math. Phys. 62, 996–1006 (2022). https://doi.org/10.1134/S0965542522060100

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  • DOI: https://doi.org/10.1134/S0965542522060100

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