On large deviations for dynamical systems randomly perturbed by a fast Markov process (Q2777849)

The method of large deviations proposed by \textit{H. Cramér} [Actuar. Sci. Indust. 736, 5-23 (1938)] as an extension of the central limit theorem was essentially generalized by \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 19, 261-286 (1966; Zbl 0147.15503) and CBMS-NSF Reg. Conf. Ser. Appl. Math. 46 (1984; Zbl 0549.60023)], who introduced the notion of large deviations property. Large deviations for randomly perturbed dynamical systems were considered by \textit{M. Freidlin} and \textit{A. Wentzell} [see, for example, ``Random perturbations of dynamical systems'' (1998; Zbl 0922.60006)]. Large deviations for Markov processes were investigated by \textit{M. D. Donsker} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 28, 1-47 (1975; Zbl 0323.60069), ibid. 28, 279-301 (1975; Zbl 0348.60031), ibid. 29, 389-461 (1976; Zbl 0348.60032) and ibid. 36, 183-212 (1983; Zbl 0512.60068)].NEWLINENEWLINENEWLINEIn this article the authors consider randomly perturbed systems determined by a differential equation in \(\mathbb R^d\) of the form \(dx_{\varepsilon}(t)/dt=a(x_{\varepsilon}(t),y_{\varepsilon}(t))\), \(x_{\varepsilon}(0)=x_0,\) where \(y_{\varepsilon}(t)=y(t/{\varepsilon})\) and \(y(t)\) is an ergodic homogeneous Markov process in a compact phase space \(Y\). Let \(\overline{x}(t)\) be a solution to the differential equation \(d\overline{x}(t)/dt=\overline{a}(\overline{x}(t))\), \(\overline{x}(0)=x_0,\) where \(\overline{a}(x)=\int a(x,y)\rho(dy)\) and \(\rho(dy)\) is the ergodic distribution of the Markov process. Methods of the theory of large deviations are applied to estimate the probability \(P\{\sup_{0\leq t\leq T}|x_{\varepsilon}(t)-\overline{x}(t)|>h\}\) of deviation of trajectories \(x_{\varepsilon}(t)\) of randomly perturbed dynamical systems from the trajectory \(\overline{x}(t)\) of the averaged system.