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We study the limiting behavior of fluctuations of synchronized system with small noise around their averaged deterministic limit. There are three regimes depending on how fast the intensity of the noise goes to zero relative to coupling parameter. By simple transformations, the original synchronized system is equivalently converted into the slow–fast system, then we derive the central limit theorem of the slow variable by the martingale method. Therefore, the fluctuation properties corresponding to the original synchronized system are derived. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a stochastic differential equation with the coefficients of class C ∞ , the successive approximations of Picard converge in the C ∞ -sense, which implies that the solution to stochastic differential equation is of class C ∞ in the initial value x almost surely. [ABSTRACT FROM AUTHOR]