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This paper focuses on systems of nonlinear second-order stochasticdifferential equations with multi-scales. The motivation stems frommathematical physics and statistical mechanics, for example, stochasticacceleration in random environment. Our main effort is to obtain averaging andlarge deviations principles. When the fast-varying process is a diffusion,neither Lipschitz continuity nor linear growth is needed. Our approach is basedon combinations of the intuition from Smoluchowski-Kramers approximation, theconcepts of relatively large deviations compactness, and identification of ratefunctions. When the fast-varying process is under a general set up with nospecified structure, we establish the large deviations principle of theunderlying system under the assumption on the local large deviations principlesof the corresponding first-order system. Some applications and numericalexamples are provided to illustrate our results.

A Class of Systems of Nonlinear Second-Order Stochastic Differential Equations: Averaging and Large Deviations Principles