## 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

Zbl 0767.60078
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