Stable dynamical systems under small perturbations
type: Item
instance of: scholarly article
title: Stable dynamical systems under small perturbations
zbMATH Open document ID: 0652.60060
DOI: 10.1016/0022-247X(87)90107-7
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: 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 \textit{M. Frejdlin} and \textit{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 \textit{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'.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 4064180
zbMATH Keywords: degenerate diffusions
zbMATH Keywords: large deviations
zbMATH Keywords: control systems
zbMATH Keywords: asymptotics of the mean exit time
zbMATH Keywords: exponential estimates of Freidlin and Venttzel'
author: Jerzy Zabczyk
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4064180