On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Let Xt be a reversible and positive recurrent diffusion in ℝd described by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{t}=x+\sigma\,b(t)+\int_{0}^{t}m(X_{s})\mathrm {d}s,$$\end{document} where the diffusion coefficient σ is a positive-definite matrix and the drift m is a smooth function. Let Xt(A) denote the image of a compact set A⊂ℝd under the stochastic flow generated by Xt. If the divergence of the drift is strictly negative, there exists a set of functions u such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim_{t\to\infty}\int_{\ensuremath {X_{t}}(A)}\ensuremath {u}(x)\mathrm {d}x=0\quad\mbox{a.s.}$$\end{document} A characterization of the functions u is provided, as well as lower and upper bounds for the exponential rate of convergence.