Freidlin - Wentzell Type Large Deviations for Smooth Processes

#### R. Liptser, V. Spokoiny, A.Yu. Veretennikov

2002, v.8, Issue 4, 611-636

ABSTRACT

We establish large deviation principle for the family of vector-valued random processes $X^\varepsilon,\varepsilon\to 0$ defined by ordinary differential equations (under $0<\kappa<1/2$) $$\dot{X}_t^\varepsilon=F(X_t^\varepsilon)+\varepsilon^{1/2-\kappa} G(X_t^\varepsilon)\dot{W}^\varepsilon_t,$$ where $\dot{W}^\varepsilon_t={\varepsilon}^{-1/2}g(\xi_{t/\varepsilon})$, $\xi_t$ is a vector-valued ergodic diffusion satisfying, so called, recurrence condition'' and $g$ is a vector-function with zero barycenter with respect to the invariant measure of $(\xi_t)$. A choice of $\kappa<1/2$ provides the rate function of Freidlin - Wentzell type.

Keywords: moderate deviations,Poisson decomposition,Puhalskiitheorem