MDP for integral functionals of fast and slow processes with averaging

Abstract

We establish large deviation principle (LDP) for the family of vector-valued random processes (X ε , Y ε ), ε → 0 defined as1991 Mathematics Subject Classification. 60J27,60F10.

Moderate deviations for systems of slow-fast diffusions

“…Let ∆ = ∆(ε) ↓ 0 as ε ↓ 0, whose role is to exploit a time-scale separation. Let A 1 , A 2 , B, and Γ be Borel (13). Associate with (X ε,u ε , Y ε,u ε ) and u ε a family of occupation measures P ε,∆ defined by…”

“…Notice that Theorem 2 of [14] has a statement for the growth only for the solution and its y−derivative. ) be the strong solution to (13). Then the infimum of the representation in (11) can be taken over all controls such that Proof.…”

“…Lemma C.1. Assume Conditions 2.1, 2.2, and 2.3 and define the processes X ε,u ε and Y ε,u ε by (13). Let N < ∞ such that almost surely sup ε>0 1 0 |u ε (s)| 2 ds < N.…”

“…In [12] the author studies the moderate deviations for (1) in the case of ε = δ with b = 0 (averaging regime) and with the fast process Y ε t being independent of the driving noise of the slow process X ε t , using different methods. In [13] the authors study the MDP for integrated functionals of systems like (1), in the averaging regime (i.e. when b = 0) and with the fast process being independent of the driving noise of the slow process.…”

“…Take expectations, and for fixed ε, all terms are bounded by Lemmas B.1, B.2, and B.3 and Doob's martingale inequality.Lemma B.5. Assume Conditions 2.1, 2.2, 2.3 and 2.4 and define the processes X ε,u ε and Y ε,u ε by(13). Let N < ∞ such that almost surely…”