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Petites perturbations aléatoires des systèmes dynamiques: développements asymptotiques. (French) Zbl 0591.60023
Let \(U\subset {\mathbb{R}}^ m\) be an open set and let \((x^{\epsilon}\); \(0<\epsilon <\epsilon_ 0)\) be a family of diffusions indexed by a ”small” parameter \(\epsilon\) and satisfying the Itô-equations: \(dx_ t^{\epsilon}=\epsilon s_ t(x_ t^{\epsilon})d\omega_ t+b_ t(\epsilon,x_ t^{\epsilon})dt,\) where \(\omega\) is the Brownian motion on \({\mathbb{R}}^ k\). The matrix and vector fields \(s_ t(x)\) and \(b_ t(\epsilon,x)\) are supposed to be sufficiently smooth and \(a_ t(x)=s_ t(x)s_ t(x)^*\) is supposed to be invertible. Let C(U) be the class of U-valued functions defined on [0,T]. It is known that for a large class of measurable sets \(A\subset C(U)\) \(\epsilon^ 2\log P(x^{\epsilon}_{0,T}\in A)\sim -\Lambda (A)=-\inf_{f\in A} \Lambda (f)\quad\) as \(\epsilon\) \(\downarrow 0\), where \(\Lambda\) is the action functional.
In this article the author proves that, for \(\partial A\) sufficiently smooth, the following precise asymptotic development holds: \[ P(x^{\epsilon}_{0,T}\in A)=(a_ 0+a_ 1\epsilon +...+a_ L\epsilon^ L+O(\epsilon^{L+\rho}))\exp (-\quad \Lambda (A)\epsilon^{-2}+\Lambda_ 1(A)\epsilon^{-1}), \] where \(\rho >0\) is determined by A and L is associated to the class of \(\partial A\) and to the class of the coefficients of \(dx^{\epsilon}\). In particular, if \(b_ t(\epsilon,x)\) does not depend on \(\epsilon\), \(\Lambda_ 1=0\) and therefore, as \(\epsilon\) \(\downarrow 0\), \(P(x^{\epsilon}_{0,T}\in A)\sim \epsilon a_ 1\exp (-\Lambda (A)\epsilon^{-2})\) if A is such that \(\Lambda\) (A)\(\neq 0\).
Reviewer: M.Dozzi

60F10 Large deviations
93E10 Estimation and detection in stochastic control theory
60G60 Random fields
93E15 Stochastic stability in control theory
70K50 Bifurcations and instability for nonlinear problems in mechanics