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We develop the large deviations principle for synchronized system with small noise. Depending on the interaction between the intensity of the noise with coupling strength, we get different behavior. By simple transformations, the original synchronized system is equivalently converted into the slow-fast system, then we derive the representations for the action functional of the slow variables via weak convergence methods. Therefore, the large deviation properties corresponding to the original synchronized system are derived. In particular, we present a large deviation principle for a particular system in view of Smoluchowski–Kramers arguments and study the synchronization of the quasipotential for a linear system. [ABSTRACT FROM AUTHOR]

In this paper, we shall prove a stochastic averaging principle for two-time-scale jump-diffusion SDEs under the non-Lipschitz coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence. [ABSTRACT FROM AUTHOR]