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.
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This paper was written while the author was visiting Northwestern University and the opinions expressed in it are those of the author alone and do not necessarily reflect the views of Merrill Lynch, its subsidiaries or affiliates.
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Baldini, M. On The Invariant Measure of a Positive Recurrent Diffusion in \({\mathbb{R}}\) . J Theor Probab 20, 65–86 (2007). https://doi.org/10.1007/s10959-006-0046-x
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DOI: https://doi.org/10.1007/s10959-006-0046-x