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A Schilder theorem for non-regular Brownian functionals. (Un théorème de Schilder pour des fonctionnelles browniennes non régulières.) (French) Zbl 0810.60025

Let \({\mathcal C}_ \alpha\) be the space of functions \(x : \langle 0,1 \rangle \to \mathbb{R}\) satisfying Hölder’s condition with parameter \(\alpha\) and let \({\mathcal C}^ 0_ \alpha\) denote the subspace of \({\mathcal C}_ \alpha\) consisting of functions \(x\) for which \[ \lim_{\delta \downarrow 0} \sup_{{| t - s | \leq \delta \atop 0 \leq t \neq s \leq 1}} {\bigl | x(t) - x(s) \bigr | \over | t - s |^ \alpha} = 0. \] If \(\varphi_ n(t) = \int^ t_ 0 \chi_ n (s)ds\), \(0 \leq t \leq 1\), where \(\{\chi_ n, n \geq 1\}\) form a CONS of Haar’s functions in \({\mathcal L}^ 2 \langle 0,1 \rangle\), then it is known that the map \(T^ \alpha\) defined by \(\eta = \{\eta_ n, n \geq 1\} \to T^ \alpha (\eta) = \sum^ \infty_{n = 1} \eta_ n n^{(1/2) - \alpha} \varphi_ n\) establishes an isomorphism between the spaces \(c_ 0\) and \({\mathcal C}^ 0_ \alpha\), and the norm \(\| x \|_ \alpha = \sup_ n | \eta_ n |\) for \(x = \sum^ \infty_{n = 1} \eta_ n n^{(1/2) - \alpha} \varphi_ n \in {\mathcal C}^ 0_ \alpha\) is equivalent to classical Hölder’s one. The asymptotic behaviour of \(E \exp \{-{1 \over 2 \varepsilon} \| f - \sqrt \varepsilon B \|_ \alpha\}\) for \(f \in {\mathcal C}^ 0_ \alpha\) and \(0 < \alpha < 1/2\) is investigated, where \(B\) is the standard Brownian motion. E.g., Theorem 1 of part II states that if \(f = \sum^ \infty_{n = 1} \eta_ n n^{(1/2) - \alpha} \varphi_ n\) is sufficiently far from the origin, i.e. \(1/2 < \sum^ \infty_{n = 1} | \eta_ n | n^{1 - 2 \alpha} < \infty\), then there exists a positive integer \(\sigma\) (dependent on \(f)\) such that \[ E \exp \left\{-{1 \over 2 \varepsilon} \bigl \| f - \sqrt \varepsilon B \bigr \|_ \alpha \right\} \sim C \varepsilon^{\sigma / 2} \exp \left \{-{I(f) \over 2 \varepsilon} \right\} \quad \text{as} \quad \varepsilon \downarrow 0, \] where \(I(f) = \inf_{\psi \in {\mathcal C}^ 0_ \alpha} [\int^ 1_ 0 \dot \psi^ 2 (s)ds + \| \psi - f \|_ \alpha] \), and \(C\) is a constant. In part III the order of magnitude for \(E \exp \{-{1 \over 2 \varepsilon} \| f - \sqrt \varepsilon B \|_ \alpha\}\) when \(\sum^ \infty_{n = 1} | \eta_ n | n^{1 - 2 \alpha} < 1/2\) is estimated. The presented theorems extend some results obtained by P. Baldi and B. Roynette [Probab. Theory Relat. Fields 93, No. 4, 457-484 (1992; Zbl 0767.60078)].

MSC:

60F10 Large deviations
60J65 Brownian motion

Citations:

Zbl 0767.60078
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