Cramér's asymptotics in systems with fast and slow motions

Abstract

This paper is devoted to the averaging principle for stochastic systems with slow and intermixing fast motions. Here we (i) prove the existence of the Cramér type asymptotics for the probabilities of large deviations from an averaged motion, which implies the central limit theorem, and (ii) develop an analytic procedure for computation of this asymptotics. We use general apparatus of superregular perturbations of fiber ergodic semigroups to investigate two systems in the same way. The first of them is a cascade in which slow motions are determined by a vector field depending both on slow and fast variables, and fast motions compose a Markov chain depending on the slow variable. The second is a process defined by a system of two stochastic differential equations.

Keywords:

• Fast and slow motions
• Averaging
• Large deviations
• Cramer asymptotics
• Superregular perturbation
• Asymptotic expansion
• Primary 60F10
• 60J60
• 60J05
• Secondary 34C29
• 47A55

Acknowledgements

Supported by INTAS Project no. 99-00559, and Belarusian Basic Research Fund.