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Short Communications

Probabilities of large deviations for randomly disturbed systems and stochastic stability

M. I. Freidlin

M. V. Lomonosov Moscow State University

Abstract: Let $x_t^\varepsilon$ be a solution of the differential equation $x^\varepsilon=b(x^\varepsilon, \varepsilon\zeta), x_0=x\in R^\gamma$. Here $\zeta_t$ is a Gaussian stochastic process, $\varepsilon$ is a small parameter. Process $x_t^\varepsilon$ may be thought of as a result of small stochastic perturbations of the system $\dot{x}=b(x,0)$. Let $O$ be a stable equilibrium point of the system, $O\in D$ (a domain in $R^\gamma$) and $\tau_D^\varepsilon=\inf\{t: x_t^\varepsilon\notin D\}$.
In the paper, the main term of $\ln\mathbf{P}\{\tau_D^\varepsilon<T\}$ as $\varepsilon\rightarrow 0$ is calculated. This term characterizes stability of point $O$ under perturbations $\varepsilon\zeta_t$ over time interval $[0, T]$.

Received: 16.02.1973

 English version: Theory of Probability and its Applications, 1974, 18:4, 779–784

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