Lyapunov functions and non-trivial stationary solutions of stochastic differential equations

Abstract

Classical conditions for the exponential stability of the zero-solution of a stochastic differential equation were formulated by Hasminskii in 1980. However, an Ito equation can have stationary solutions that are (non-trivial) processes. Conditions that allow the detection of such exponentially stable non-trivial stationary solutions are formulated. These conditions are based on Lyapunov functions for the generator of the two point motion of a stochastic differential equation. Another tool that will be used is the pullback-convergence technique for random dynamical systems.