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

Michele Baldini ()
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Michele Baldini: Global Equity Linked Products

Journal of Theoretical Probability, 2007, vol. 20, issue 1, 65-86

Abstract: 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.

Keywords: Stochastic flow; diffusion; diffeomorphism; invariant measure; 60J60; 60H15; 37A50 (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-006-0046-x

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