On random perturbed mechanical systems with two degrees of freedom (Q2722271)

The author considers differential equations NEWLINE\[NEWLINEdx_{\varepsilon}(t)=a(x_{\varepsilon}(t),y(t/\varepsilon))NEWLINE\]NEWLINE where \(x_{\varepsilon}(t)\) is an \(R^{d}\)-valued stochastic process, \(y(t)\) is a homogeneous Markov process in a measurable space \(Y\), \(a(x,y)\) is a measurable function from \(R^{d}\times Y\) into \(R^{d}\) which is smooth enough in \(x,\) \(\varepsilon>0\) is a small parameter. NEWLINENEWLINENEWLINEAsymptotic properties of the stochastic process \(x_{\varepsilon}(t)\) are investigated as \(\varepsilon\to 0.\) The diffusion approximation for the energy of a two-dimensional above-mentioned mechanical system randomly perturbed by a fast homogeneous ergodic Markov process \(y(t/\varepsilon)\) is considered. It is proved that the energy converges weakly to a diffusion process on the graph generated by the orbits of the averaged system.