On the behaviour of expectation of the energy of stochastic harmonic oscillator (Q2755241)

This paper deals with the harmonic oscillator with friction described by the stochastic differential equation \(dx(t)=x(t)d\xi(t),\;x_1(0)=u_0, x_2(0)=\dot u_0\), where \(x(t)=(x_1(t),x_2(t))\); \(\xi(t)=\int_{0}^{t}B_{q(s)}ds+ \int_{0}^{t}B_{g(s)}dw(s)\); \(w(t)\) is a Wiener process; the form of matrices \(B_{f}, f=q,g\) depends on the cases: \(|h |<k\), \(|h |>k\), \(|h |=k\); \(k>0\) and \(h\) are the oscillator parameters; \(2h\) is the friction coefficient. The author considers the case when the random perturbations act at an angle to the vector of the phase velocity. The exact forms of \(E\xi_{i}(t), E\xi_{i}(t)\xi_{i}(s), i=1,2, E\xi_{1}(t)\xi_{2}(s)\) and \(E[(\xi_{2}^2(s)+k^2\xi_{1}^2(s))/2]\) are obtained and the limit behaviour of these values as \(t\to\infty\) are investigated.