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A note on conditional exponential moments and Onsager-Machlup functionals. (English) Zbl 0756.60038

Let \(\Phi\in L^ 2[0,1]\) be a given deterministic function, let \(\omega_ t\) be a standard Brownian motion, and define \(I(\Phi) \widehat= \int_ 0^ 1\Phi_ t d\omega_ t\). The authors prove that \[ E\left(\exp\int_ 0^ 1\Phi(t)d\omega_ t \mid\;\|\omega\|<\varepsilon\right) \to 1 \qquad\text{as } \varepsilon\to 0, \] where \(\|\cdot\|\) is any “reasonable” norm on \(C_ 0[0,1]\). The point of start is a previous result of O. Zeitouni by virtue of which, if for a given deterministic path \(\Phi_ t\), \(t\in[0,1]\), one has that \[ E(\exp I(\Phi)\mid |||\omega|||<\varepsilon)\to 1 \qquad\text{as } \varepsilon\to 0, \] where \(|||\cdot|||\) is the supremum norm on \([0,1]\), then the Onsager-Machlup functional for \(\hat\Phi_ t=\int_ 0^ t\Phi_ t dt\) follows, that is, if \(x_ t\) is a diffusion satisfying \(dx_ t=f(x_ t)dt+d\omega_ t\), with \(f\) bounded and having two continuous bounded derivatives, then \[ {{\text{Prob}\{||| x-\hat\Phi|||<\varepsilon\}} \over {\text{Prob}\{|||\omega|||<\varepsilon\}}}\to\exp \left\{-{1\over2}\int_ 0^ 1[(\Phi_ t-f(\hat\Phi_ t))^ 2+f'(\hat\Phi_ t)]dt\right\} \qquad\text{as }\varepsilon\to 0. \] Applications to the computation of Onsager-Machlup functionals are emphasized, too.
Reviewer: G.Orman (Braşov)

MSC:

60G15 Gaussian processes
60F10 Large deviations
60J65 Brownian motion
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