Asymptotics of superregular perturbations of fiber ergodic semigroups

Abstract

In this paper, we investigate a semigroup of conditional expectation operators A_{ \epsilon }^{t}, generated by a stochastic system with slow and intermixing fast motions, in which the slow motions have a speed of order ε. This semigroup, considered as perturbation of A_{0}^{t}, possess a number of unexpected properties; the most important of them is superregularity. First we study these properties. Then we construct an asymptotic expansion for the family A_{ \epsilon }^{t/ \epsilon }e^{ \mgreek{x} F/ \epsilon } by powers of ξ, ε, where ξ is a small complex parameter and F is a function of the slow variable. We reveal a new non-trivial phenomenon: each coefficient of the last expansion appears as a sum of four terms of different types. This expansion gives a powerful tool for proving some probability limit theorems for the slow motion behavior over the time periods of order ε ^-1.

Keywords:

• Semigroup
• Superregular perturbation
• Fast and slow motions
• Asymptotic expansion
• Large deviations
• Primary 47A55
• Secondary 60H10
• 60J05

Acknowledgements

Supported by Belarusian Basic Research Fund and by INTAS project No. 99-00559.