A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

Previous article Next article A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand SidesR. Z. Khas’minskiiR. Z. Khas’minskiihttps://doi.org/10.1137/1111038PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. Z. Khas'minskii, On stochastic processes defined by differential equations with a small parameter, Theory Prob. Applications, 11 (1966), 211–228, (English translation) 10.1137/1111018 LinkGoogle Scholar[2] R. L. Stratonovich, Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York, 1963xiii+292, Vol. II, (1967), (Translated by R. L. Silverman.) MR0158437 Google Scholar[3] R. L. Stratonovich, Masters Thesis, Conditional Markov Processes and Their Application to Optimal Control Theory, Doctoral Dissertation, Mosc. Univ., Moscow, 1965, (In Russian.) Google Scholar[4] I. A. Ibragimov, Some limit theorems for stochastic processes stationary in the strict sense, Dokl. Akad. Nauk SSSR, 125 (1959), 711–714, (In Russian.) MR0105180 0087.13302 Google Scholar[5] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Applications, 7 (1962), 349–382, (English translation.) 10.1137/1107036 0119.14204 LinkGoogle Scholar[6] R. L. Dobrushin, The central limit theorem for nonstationary Markov chains I, II, Theory Prob. Applications, 1 (1956), 65–80; 329–383, (English translation.) Google Scholar[7] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[8] V. A. Volkonskii˘ and , Yu. A. Rozanov, Some limit theorems for random functions. I, Theor. Probability Appl., 4 (1959), 178–197, (English translation.) MR0121856 0092.33502 LinkGoogle Scholar[9] A. V. Skorokhod, Studies in the theory of random processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965viii+199 MR0185620 0146.37701 Google Scholar[10] Yu. V. Prokhorov, Convergence of stochastic processes and limit theorems in probability theory, Theory Prob. Applications, 1 (1956), 157–214, (English translation.) 10.1137/1101016 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Averaging of neutral stochastic partial functional differential equations involving delayed impulses25 May 2021 | Applicable Analysis, Vol. 101, No. 18 Cross Ref The averaging method for doubly perturbed distribution dependent SDEsStatistics & Probability Letters, Vol. 189 Cross Ref Stochastic analysis of a nonlinear energy harvester with fractional derivative damping11 March 2022 | Nonlinear Dynamics, Vol. 108, No. 3 Cross Ref Probabilistic analysis of drill string under uncertain conditions for flow velocity and axial compressive loadApplied Ocean Research, Vol. 122 Cross Ref The applicability of stochastic averaging method to solve the ship rolling response excited by narrow-band wavesOcean Engineering, Vol. 251 Cross Ref A mechanism for the skew of ensemble forecasts15 March 2022 | Quarterly Journal of the Royal Meteorological Society, Vol. 148, No. 744 Cross Ref Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments13 January 2022 | Nonlinear Dynamics, Vol. 107, No. 4 Cross Ref Improved stochastic averaging method using Hamiltonian for parametric rolling in irregular longitudinal waves25 May 2021 | Journal of Marine Science and Technology, Vol. 27, No. 1 Cross Ref Averaging Principles for Stochastic 2D Navier–Stokes Equations16 January 2022 | Journal of Statistical Physics, Vol. 186, No. 2 Cross Ref Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle12 February 2022 | Fractal and Fractional, Vol. 6, No. 2 Cross Ref Response and reliability analysis of random time-delay controlled systems undergo wide-band random excitationsProbabilistic Engineering Mechanics, Vol. 67 Cross Ref Stochastic Hopf Bifurcation of MDOF Quasi-Integrable Hamiltonian Systems with Delayed Feedback Fractional-Order PD Controller2 September 2021 | International Journal of Structural Stability and Dynamics, Vol. 21, No. 13 Cross Ref A novel method for evaluating the averaged drift and diffusion coefficients of high DOF quasi-non-integrable Hamiltonian systems10 November 2021 | Nonlinear Dynamics, Vol. 106, No. 4 Cross Ref Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulses16 September 2020 | Stochastics, Vol. 93, No. 6 Cross Ref Amplitude-phase description of stochastic neural oscillators across the Hopf bifurcation9 July 2021 | Physical Review Research, Vol. 3, No. 3 Cross Ref Moment stability of viscoelastic system influenced by non-Gaussian colored noiseJournal of Sound and Vibration, Vol. 502 Cross Ref Stationary response of colored noise excited vibro-impact system*Chinese Physics B, Vol. 30, No. 6 Cross Ref Modeling of the quasi-periodic galloping response type under combined harmonic and random excitationComputers & Structures, Vol. 247 Cross Ref Averaging Principle for Backward Stochastic Differential EquationsDiscrete Dynamics in Nature and Society, Vol. 2021 Cross Ref Diffusion Approximation for Noise-Induced Evolution of First Integrals in Multifrequency Systems23 February 2021 | Journal of Statistical Physics, Vol. 182, No. 3 Cross Ref Quantum Behavior of a Classical Particle Subject to a Random Force1 February 2021 | Foundations of Physics, Vol. 51, No. 1 Cross Ref Stochastic averaging for estimating a ship roll in random longitudinal or oblique wavesMarine Structures, Vol. 75 Cross Ref Perturbed uncertain differential equations and perturbed reflected canonical processAIMS Mathematics, Vol. 6, No. 9 Cross Ref Homogenization for stochastic reaction-diffusion equations with singular perturbation termDiscrete & Continuous Dynamical Systems - 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