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Freidlin-Wentzell large deviations in Hölder norm. (Grandes déviations de Freidlin-Wentzell en norme Höldérienne.) (French) Zbl 0811.60019

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXVIII. Berlin: Springer-Verlag. Lect. Notes Math. 1583, 293-299 (1994).
The authors prove that the Freidlin-Wentzell large deviation principle for small perturbations of dynamical systems can be extended to the Hölder topology of index \(\alpha\) for all \(0<\alpha<1/2\). More precisely if for all functions \(w: [0,1]\to \mathbb{R}^ d\) the Hölder norm of index \(\alpha\) is defined by \[ \| w\|_ \alpha= \sup_{0\leq s\leq t\leq 1} {{| w(s)- w(t)|} \over {| t-s|^ \alpha}}, \] then if \(0<\alpha <1/2\), for all \(x\in \mathbb{R}^ d\) and for all Borel sets of \(C_ x([0,1], \mathbb{R}^ d)\): \[ -\Lambda (\overset {\circ} {A})\leq \liminf_{\varepsilon\to 0} \varepsilon^ 2 \text{Log } P(X_ \varepsilon^ x \in A)\leq \limsup_{\varepsilon\to 0} \varepsilon^ 2 \text{Log } P(X_ \varepsilon^ x\in A)\leq - \Lambda( \overline{A}), \] \(\overset {\circ} {A}\) and \(\overline {A}\) are respectively the interior and the closure of \(A\) for the Hölder topology of index \(\alpha\) and \[ \Lambda(A)= \inf \bigl( {\textstyle {1\over 2}} | h|^ 2;\;h\in H, \varphi^ x(h)\in A\bigr), \]
\[ X_ \varepsilon^ x(t)= x+\varepsilon \int_ 0^ t \sigma (X_ \varepsilon^ x (s)) dw(s)+ \int_ 0^ t b_ \varepsilon (X_ \varepsilon^ x (s))ds, \qquad 0\leq t\leq 1, \] \(b\) is the uniform limit of \(b_ \varepsilon\), and \(\varphi^ x(h)\) is the solution of the ordinary differential equation \[ \varphi^ x(h)(t)= x+\int_ 0^ t \sigma(\varphi^ x (h)(s)) \dot h(s)ds+ \int_ 0^ t b(\varphi^ x(h) (s))ds, \qquad 0\leq t\leq 1. \]
For the entire collection see [Zbl 0797.00020].

MSC:

60F10 Large deviations
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