A formula for the lyapunov numbers of a stochastic flow. application to a perturbation theorem

Abstract

We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system. (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow). This formula is analogous to that of Khas'minskii, who deals with a linear system. We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (i.e. all the exponents are negative) then so are certain stochastic perturbations of it.