Stable dynamical systems under small perturbations
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'.
exponential estimates of Freidlin and Venttzel'
scientific article; zbMATH DE number 4064180