We consider multiscale stochastic dynamical systems. In this article an intermediate reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of O(ǫ). Second, averaging derives an autonomous deterministic system up to errors of O(√ǫ). Then an intermedi...

We study the averaging-based distributed optimization solvers over random networks. show a general result on convergence of such schemes using weight-matrices that are row-stochastic almost surely and column-stochastic in expectation for broad class dependent weight-matrix sequences. In addition to implying many previously known results this domain, our work shows robustness link-failure. Also,...

<p style='text-indent:20px;'>In contrast to existing works on stochastic averaging finite intervals, we establish an principle the whole real axis, i.e. so-called second Bogolyubov theorem, for semilinear ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost automorphic etc) coefficients. Under some appropriate conditions pro...