Journal of Theoretical Probability

, Volume 20, Issue 1, pp 65–86 | Cite as

On The Invariant Measure of a Positive Recurrent Diffusion in \({\mathbb{R}}\)


Given a one-dimensional positive recurrent diffusion governed by the Stratonovich SDE \({X_t=x+\int_0^t\sigma(X_s)\bullet\hbox{d}b(s)+\int_0^t m(X_s)\hbox{d}s}\) , we show that the associated stochastic flow of diffeomorphisms focuses as fast as \({exp (-2t\int_{\mathbb{R}}\frac{m^2}{\sigma^2} d\Pi)}\) , 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.


Stochastic flow diffusion diffeomorphism invariant measure 

Subject Classification Codes

60J60 60H15 37A50 


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Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Global Equity Linked ProductsNew YorkUSA

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