## The Annals of Probability

### On large deviations of coupled diffusions with time scale separation

Anatolii A. Puhalskii

#### Abstract

We consider two Itô equations that evolve on different time scales. The equations are fully coupled in the sense that all of the coefficients may depend on both the “slow” and the “fast” variables and the diffusion terms may be correlated. The diffusion term in the slow process is small. A large deviation principle is obtained for the joint distribution of the slow process and of the empirical process of the fast variable. By projecting on the slow and fast variables, we arrive at new results on large deviations in the averaging framework and on large deviations of the empirical measures of ergodic diffusions, respectively. The proof relies on the property that an exponentially tight sequence of probability measures on a metric space is large deviation relatively compact. The identification of the large deviation rate function is accomplished by analyzing the large deviation limit of an exponential martingale.

#### Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 3111-3186.

Dates
Revised: May 2015
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.aop/1470139161

Digital Object Identifier
doi:10.1214/15-AOP1043

Mathematical Reviews number (MathSciNet)
MR3531687

Zentralblatt MATH identifier
1356.60047

#### Citation

Puhalskii, Anatolii A. On large deviations of coupled diffusions with time scale separation. Ann. Probab. 44 (2016), no. 4, 3111--3186. doi:10.1214/15-AOP1043. https://projecteuclid.org/euclid.aop/1470139161

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