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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 281–295 (Mi tvp2529)

Some problems concerning stability under small stochastic perturbations

A. D. Venttsel', M. I. Freidlin

Moscow

Abstract: Let $x_0$ be a stable equilibrium point of a dynamic system $\dot x=b(x)$ in $R^r$; a Markov stochastic process $x_t^\varepsilon$ appears as a result of small stochastic perturbations of this system: $dx_t^\varepsilon=b(x_t^\varepsilon) dt+\varepsilon\sigma(x_t^\varepsilon) d\xi_t$ where $\xi_t$ is a Wiener process. Problems concerning stability of the point $x_0$ with respect to the stochastic process $x_t^\varepsilon$ are considered.
All trajectories of the process $x_t^\varepsilon$ sooner or later, leave each neighbourhood of the equilibrium point. The problem arises how to choose a region of a given area for which the mean exit time is maximum? Another problem setting: suppose that one can control the process $x_t^\varepsilon$ by chosing a drift vector $b(x)$ at each point $x$ of some set of vectors $\Pi(x)$. How should one control the process so that the mean exit time of a given region would be maximum (minimum)? Asymptotically optimal solutions to these questions are given: the control proposed by the authors is not worse (not essentially worse) than any other control if $\varepsilon$ is sufficiently small; the mean exit time of any other region $G$ of a given area is less than that of the region the authors point at if $\varepsilon$ is small.
The way of solving these problems is to estimate the main term of the mean exit time of a given region $G$ when $\varepsilon\to0$. This main term is $\exp\{\frac1{2\varepsilon^2}\min\limits_{y\in\partial G}V(x_0,y)\}$ where $V(x_0,x)$ is a function that does not depend on the region and can be found as a solution of a specific problem for the differential equation
$$\sum a^{ij}(x)\frac{\partial V}{\partial x^i}\frac{\partial V}{\partial x^j}+4\sum b^i(x)\frac{\partial V}{\partial x^i}=0,\quad(a^{ij}(x))=\sigma(x)\sigma^*(x).$$
In order to solve the optimal control problem, a non-linear partial differential equation is considered. In the case of shift-invariance this equation can be solved by means of a certain geometrical construction.

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English version:
Theory of Probability and its Applications, 1973, 17:2, 269–283

Bibliographic databases:

Citation: A. D. Venttsel', M. I. Freidlin, “Some problems concerning stability under small stochastic perturbations”, Teor. Veroyatnost. i Primenen., 17:2 (1972), 281–295; Theory Probab. Appl., 17:2 (1973), 269–283

Citation in format AMSBIB
\Bibitem{VenFre72} \by A.~D.~Venttsel', M.~I.~Freidlin \paper Some problems concerning stability under small stochastic perturbations \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 2 \pages 281--295 \mathnet{http://mi.mathnet.ru/tvp2529} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=298155} \zmath{https://zbmath.org/?q=an:0268.93032} \transl \jour Theory Probab. Appl. \yr 1973 \vol 17 \issue 2 \pages 269--283 \crossref{https://doi.org/10.1137/1117031} 

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• http://mi.mathnet.ru/eng/tvp/v17/i2/p281

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This publication is cited in the following articles:
1. M. I. Freidlin, “The averaging principle and theorems on large deviations”, Russian Math. Surveys, 33:5 (1978), 117–176