Asymptotic behavior of a system of stochastic equations (Q2784974)

The authors consider the stochastic equation NEWLINE\[NEWLINEdx_{\varepsilon}(t)=a(x_{\varepsilon}(t)) dt+\sigma(x_{\varepsilon}(t)) \varepsilon^{-1/2}\eta(t/\varepsilon) dt, \quad x_{\varepsilon}(0)=0,NEWLINE\]NEWLINE where \(\varepsilon\) is a small parameter and \(\eta(t)\) is a weakly stationary stochastic process with zero mean satisfying uniformly strong mixing condition with mixing coefficient \(\varphi(s)\to 0\), as \(s\to\infty\). The rate of convergence of solution \(x_{\varepsilon}(t)\) of the considered equation to a solution of the diffusion equation is studied. The asymptotic behavior of solutions of the equation \(dx(t)=\sigma(x(t))\eta(t)dt,\;x(0)=0\), where \(\sigma(x)\) is a periodic function with period 1, is investigated. These results are used to prove the law of iterated logarithm in the Strassen form for solutions of the last equation.