Markov Perturbations on Large Time Intervals

Abstract

In this chapter we consider families of diffusion processes $(X_t^\varepsilon,\,{\text{P}}_x^\varepsilon)$ on a connected manifold M. We shall assume that these families satisfy the hypotheses of Theorem 3.2 of Ch. 5 and the behavior of probabilities of large deviations from the “most probable” trajectory—the trajectory of the dynamical system $\dot x_t \, = \,b(x_t )$ —can be described as ε → 0, by the action functional $\varepsilon ^{ - 2} S\left( \varphi \right)\, = \,\varepsilon ^{ - 2} S_{T_1 T_2 } \left( \varphi \right),$ where

$$ S(\varphi )\, = \,\frac{1} {2}\int_{T_1 }^{T_2 } {\mathop \sum \limits_{ij} } \,a_{ij} (\varphi _t )(\dot \varphi _t^i \, - \,b^i (\varphi _t ))(\dot \varphi _t^j \, - \,b^j (\varphi _t ))\,dt. $$