Phase picture of the diffusion processes with the degenerate diffusion matrices

Abstract

Nonlinear stochastic differential Ito's equations in R n with the degenerate diffusion matrices are considered. Methods are proposed for investigation of the locally invariant sets of SDE's. Classes of SDE's which have given locally invariant set are constructed.

On stability of sets for sequences of random polygonal lines

“…The nonsingularity of the diffusion matrix of the limit equation is an essential condition. At the same time, the diffusion matrix of the Itô stochastic differential equation that has invariant curves [3,4] is singular on these curves. In the present paper, we investigate the convergence of sequences of random polygonal lines to sets that, in particular, can be invariant for solutions of stochastic differential equations.…”