  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.
