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Stable dynamical systems under small perturbations. (English) Zbl 0652.60060

Let \(X^{x,\epsilon}\) be a solution of the stochastic Ito equation \[ dX=f(X)dt+\epsilon a(X)dW_ t,\quad X(0)=x, \] and D an open set containing an equilibrium point \(\hat x\) for the equation \(\dot z=f(z)\). The paper is concerned with the (exit) problem of finding the asymptotics of the mean exit time (from D) and the distribution of the exit place as \(\epsilon\) \(\downarrow 0.\)
Earlier results of M. Frejdlin and A. Venttzel’ [see “Random perturbations of dynamical systems.” (1984; Zbl 0522.60055)], are extended to the degenerate case when the matrix a(x)a(x) T, \(x\in R\) n, is, in general, singular. The main technical tool is R. G. Azencott’s generalization [see École d’été de probabilités de Saint-Flour VIII-1978, Lect. Notes Math. 774, 1-176 (1980; Zbl 0435.60028)] of the exponential estimates of Freidlin and Venttzel’.
Reviewer: J.Zabozyk

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F10 Large deviations
93E15 Stochastic stability in control theory
60J60 Diffusion processes
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