On the invariant measure of a positive recurrent diffusion in R

Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE $$X_t=x+\int_0^t\sigma(X_s)\strat db(s)+\int_0^t m(X_s) ds,$$ we show that the associated stochastic flow of diffeomorphisms focuses as fast as $\mathrm{exp}(-2t\int_{R}\frac{m^2}{\sigma^2} d\Pi)$, where $d\Pi$ is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is $d\Pi$. Applications to stationary solutions of $X_t$, asymptotic behavior of solutions of SPDEs and random attractors are offered.