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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

##### MSC:
 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