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Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 444–462 (Mi tvp640)  

This article is cited in 6 scientific papers (total in 6 papers)

Предельная теорема для решений дифференциальных уравнений со случайной правой частью

R. Z. Khas'minskii

Moscow

Abstract: The asymptotic behaviour of the solution $X_\varepsilon(t,\omega)$ of equation (0.1) as $\varepsilon\to0$ is considered. The main assumptions are the following ones: 1) condition (1.1) is fulfilled and the processes $F^{(i)}(x,t,\omega)$ satisfy Ibragirnov's mixing condition (1.5) with $T^6\beta(T)\downarrow0$ as $T\to\infty$, 2) limits (1.4) exist and $\overline\Phi^0(x)\equiv0$. The weak convergence of the process $X_\varepsilon(\tau,\omega)$ $(\tau=\varepsilon^2t)$ to a Markov process $X_0(\tau,\omega)$ is proved. Moreover the local characteristics of the process $X_0(\tau,\omega)$ are calculated. An application of this theorem to the problem of parametric excitation of linear systems by random forces is considered

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English version:
Theory of Probability and its Applications, 1966, 11:3, 390–406

Bibliographic databases:

Received: 14.09.1965

Citation: R. Z. Khas'minskii, “Предельная теорема для решений дифференциальных уравнений со случайной правой частью”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 444–462; Theory Probab. Appl., 11:3 (1966), 390–406

Citation in format AMSBIB
\Bibitem{Kha66}
\by R.~Z.~Khas'minskii
\paper Предельная теорема для решений дифференциальных уравнений со случайной правой частью
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 3
\pages 444--462
\mathnet{http://mi.mathnet.ru/tvp640}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=203789}
\zmath{https://zbmath.org/?q=an:0202.48601}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 3
\pages 390--406
\crossref{https://doi.org/10.1137/1111038}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Sarafyan, R. G. Sarafyan, M. I. Freidlin, “Degenerate diffusion processes and differential equations with a small parameter”, Russian Math. Surveys, 33:6 (1978), 257–258  mathnet  crossref  mathscinet  zmath
    2. Yu. N. Barabanenkov, “Cauchy problem for stochastic Liouville equation with randomly variable Hamiltonian of perturbations in the form of a bounded operator”, Theoret. and Math. Phys., 42:1 (1980), 66–73  mathnet  crossref  mathscinet  zmath  isi
    3. S. M. Pergamenshchikov, “Asymptotic expansions for a model with distinguished “fast” and “slow” variables, described by a system of singularly perturbed stochastic differential equations”, Russian Math. Surveys, 49:4 (1994), 1–44  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Pardoux E., Veretennikov A.Y., “On the Poisson equation and diffusion approximation 3”, Annals of Probability, 33:3 (2005), 1111–1133  crossref  mathscinet  zmath  isi
    5. Givon D., Kevrekidis I.G., Kupferman R., “Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems”, Communications in Mathematical Sciences, 4:4 (2006), 707–729  mathscinet  zmath  isi
    6. Medvedev A.V., “Dinamika bystro vraschayuschegosya nekontaktnogo giroskopa pri ploskikh sluchainykh vibratsiyakh tochki podvesa”, Vestnik tambovskogo universiteta. seriya: estestvennye i tekhnicheskie nauki, 18:1 (2013), 70–72  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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