On the invariant measure of a positive recurrent diffusion in R

Abstract

Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE \[ X_t=x+\int_0^tσ(X_s)\strat db(s)+\int_0^t m(X_s) ds, \] we show that the associated stochastic flow of diffeomorphisms focuses as fast as $ \mathrm{exp}(-2t\int_{R}\frac{m^2}{σ^2} dΠ)$, where $dΠ$ 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 $dΠ$. Applications to stationary solutions of $X_t$, asymptotic behavior of solutions of SPDEs and random attractors are offered.