On randomly perturbed linear oscillating mechanical systems (Q2722160)

Differential equations with random functions containing a small parameter were first studied by \textit{R. Z. Khas'minskij} [Theory Probab. Appl. 11, 211-228 (1966), translation from Teor. Veroyatn. Primen. 11, 240-259 (1966; Zbl 0168.16002); and ibid. 11, 390-406 (1966), resp. ibid. 11, 444-462 (1966; Zbl 0202.48601)]. The problem considered in the first article is related to diffusion approximation for randomly perturbed differential equations. It is proved that the amplitudes and the phases of eigenoscillations of a linear oscillating system perturbed by either a fast Markov process or small Wiener process can be described asymptotically as a diffusion process whose generator is calculated. Under various conditions problems of such kind were studied by \textit{R. Z. Khas'minskij} [Kybernetika, Praha 4, 260-279 (1968; Zbl 0231.60045)], \textit{G. C. Papanicolaou, D. Stroock} and \textit{S. R. S. Varadhan} [in: 1976 Duke Turbul. Conf., Durham 1976, VI.1--VI.120 (1977; Zbl 0387.60067)], the author [``Asymptotic methods in the theory of stochastic differential equations'' (1989; Zbl 0695.60055)], \textit{M. I. Freidlin} and \textit{A. D. Wentzell} [Mem. Am. Math. Soc. 523 (1994; Zbl 0804.60070)]. The same problems have been studied by \textit{V. S. Korolyuk} and the reviewer [``Semi-Markov random evolutions'' (1994; Zbl 0813.60083) and ``Evolution of systems in random media'' (Boca Raton, 1995)], where results of applications of the general limit theorems for random evolutions to the stochastic evolutionary systems in random media, in particular, to dynamical systems under random perturbations (Markov and semi-Markov) are given.