On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Let $X_t$ be a reversible and positive recurrent diffusion in $R^d$ described by \begin{equation}\nonumber X_t=x+\sigma b(t)+\int_0^tm(X_s)\dif s, \end{equation} 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 $A\subset R^d$ 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 $$\lim_{t\to\infty} \int_{X_t(A)}u(x)\dif x=0\quad{a.s.}$$ A characterization of the functions $u$ is provided, as well as lower and upper bounds for the exponential rate of convergence.