On the negativity of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion

We provide sign information for the top Lyapunov exponent for a stochastic differential equation driven by fractional Brownian motion. To this aim we analyze the stochastic dynamical system generated by such an equation, obtain a random dynamical system and construct an appropriate invariant measure. Suitable estimates for its density together with Birkhoff’s ergodic theorem imply the negativity of the top Lyapunov exponent by increasing the noise intensity.