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The Laplace and the stationary phase methods on Wiener space. (Méthode de Laplace et de la phase stationnaire sur l’espace de Wiener.) (French) Zbl 0666.60026

Let \(X^{\varepsilon}\) be a diffusion process with generator \((\varepsilon^ 2/2)\sum a_{ij}(x)\partial^ 2_{ij}+\sum b_ i(\varepsilon,x)\partial_ i\) and \(f, F, G\) be sufficiently smooth functionals on Wiener space. Consider the functional \[ I(\varepsilon)=E[f(X^{\varepsilon})\exp (-(1/\varepsilon^ 2)(F+iG)(X^{\varepsilon}))]. \] For the elliptic case and \(G\equiv 0\) R. Azencott [Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Lect. Notes Math. 921, 237–285 (1982; Zbl 0484.60064)] gave an asymptotic expansion for \(\varepsilon\) \(\to 0.\)
In the present paper this result is generalized to degenerate diffusions. The leading term is of the form \(\alpha_ 0e^{-a/\varepsilon^ 2- c/\varepsilon}\) and an explicit formula for \(\alpha_ 0\) is given. Using Malliavin calculus the author is able to treat also the case of \(G\not\equiv 0\).

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60F10 Large deviations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60J60 Diffusion processes

Citations:

Zbl 0484.60064
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References:

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