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Teor. Veroyatnost. i Primenen.:

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

This article is cited in 1 scientific paper (total in 1 paper)

Some problems concerning stability under small stochastic perturbations

A. D. Venttsel', M. I. Freidlin


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

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Received: 23.07.1970

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
\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
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 2
\pages 269--283

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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  mathnet  crossref  mathscinet  zmath
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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