On large deviations in the averaging principle for SDE's with a ``full dependence'', revisited (Q1943313)

The paper is a correction of \textit{A. Yu. Veretennikov} [Ann. Probab. 27, No. 1, 284--296 (1999; Zbl 0939.60012)]. Here, the author considers a slow-fast system of the type \[ d X^\varepsilon_t = f(X^\varepsilon_t, Y^\varepsilon_t) dt, \qquad dY^\varepsilon_t = \varepsilon^{-2} B(X^\varepsilon_t, Y^\varepsilon_t) dt + \varepsilon^{-1} C(X^\varepsilon_t, Y^\varepsilon_t) dW_t, \] where \(\varepsilon \to 0\). In the limit, \(\bar X = \lim_\varepsilon X^\varepsilon\) solves an averaged equation of the type \(d \bar X_t = \bar f (\bar X_t) dt\), and here the large deviations around that limit are studied. In the original paper there was a gap in the argument, which is fixed in this correction, and the arguments are overall simplified.