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On large deviations in the averaging principle for SDEs with a “full dependence”. (English) Zbl 0939.60012

Ann. Probab. 27, No. 1, 284-296 (1999); correction Discrete Contin. Dyn. Syst., Ser. B 18, No. 2, 523-549 (2013).
Large deviations in the averaging principle are established for the system \(Z_t = (X_t,Y_t), X_t \in E^d, Y_t \in M \), described by \( dX_t = f(Z_t) dt\), \(dY_t = \varepsilon^{-2} B(Z_t)dt + \varepsilon^{-1} C(Z_t)dW_t\), \(Z_0 =(x_0,y_0)\), where \(M\) is a compact manifold of dimension \(l\). The author’s approach is based on using two different scales of partitions of \([0,T]\), the one by points independent of \(\varepsilon\), and the other by points dependent on \( \varepsilon^2 o( \log \varepsilon^{-1})\). In the proof of the main theorem, the author’s comments on some functions introduced by M. I. Freidlin and A. D. Wentzell [“Random perturbations of dynamical systems” (1984; Zbl 0922.60006), Section 7.5] are presented, too.
Reviewer: T.N.Pham (Hanoi)

MSC:

60F10 Large deviations
60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
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