On the Lyapunov exponent of a multidimensional stochastic flow

Let X-t be a reversible and positive recurrent diffusion in R-d described by X-t = x + sigma b(t) + integral(t)(0) m(X-s)ds, where the diffusion coefficient sigma is a positive-definite matrix and the drift m is a smooth function. Let X (t) (A) denote the image of a compact set A subset of R-d under the stochastic flow generated by X-t . If the divergence of the drift is strictly negative, there exists a set of functions u such that [GRAPHICS] A characterization of the functions u is provided, as well as lower and upper bounds for the exponential rate of convergence.