Zabczyk, J. Stable dynamical systems under small perturbations. (English) Zbl 0652.60060 J. Math. Anal. Appl. 125, 568-588 (1987). 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 Keywords:degenerate diffusions; large deviations; control systems; asymptotics of the mean exit time; exponential estimates of Freidlin and Venttzel’ Citations:Zbl 0522.60055; Zbl 0435.60028 PDFBibTeX XMLCite \textit{J. Zabczyk}, J. Math. Anal. Appl. 125, 568--588 (1987; Zbl 0652.60060) Full Text: DOI References: [1] Azencott, R. G., Sur les grand deriations, (Ecole d’Eté sz Probabilité, Sain Flour. Ecole d’Eté sz Probabilité, Sain Flour, Lecture Notes in Mathematics, Vol. 774 (1978), Springer-Verlag: Springer-Verlag New York/Berlin) [2] M. Chaleyat-Maurel; M. Chaleyat-Maurel [3] C. L. De Marco and A. R. Bergen; C. L. De Marco and A. R. Bergen [4] Faris, W. G.; Lasinio, G. Jona, Large fluctuations for nonlinear heat equation with noise, J. Physics A: Math. Gen., 15, 3025-3055 (1982) · Zbl 0496.60060 [5] Freidlin, M.; Wentzell, A., On small random perturbations of dynamical systems, Russian Math Surveys, 25, 1-55 (1970) · Zbl 0297.34053 [6] Freidlin, M.; Wentzell, A., Random Perturbations of Dynamical Systems (1984), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0522.60055 [7] Gromov, M., Structures Métriques pour les variétés riémannienes (1981), Cedic/Fernand Nathan · Zbl 0509.53034 [8] Lee, E. B.; Markus, L., Foundation of Optimal Control Theory (1967), Weley · Zbl 0159.13201 [9] Ł. Stettner; Ł. Stettner · Zbl 0681.60030 [10] Stroock, D. W., An Introduction to Large Deviations (1984), Springer-Verlag · Zbl 0552.60022 [11] Varadham, S. R.S, Large Deviations and Applications, (CBMS Lecture Notes (1983), Courant Institute of Mathematical Sciences: Courant Institute of Mathematical Sciences Washington, D.C) [12] Zabczyk, J., Exit problem and control theory, Systems Control Lett., 6, 165-172 (1985) · Zbl 0591.60050 [13] Zabczyk, J., Exit problem for infinite dimensional systems, (Da Prato, G.; Tubaro, L., Proceedings of a workshop on Stochastic PDE’s and Applications. Proceedings of a workshop on Stochastic PDE’s and Applications, Trento (1985), LNiM), to appear · Zbl 0617.60057 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.