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

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  • Michele Baldini

    (Global Equity Linked Products)

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.

Suggested Citation

  • Michele Baldini, 2007. "On The Invariant Measure of a Positive Recurrent Diffusion in $${\mathbb{R}}$$," Journal of Theoretical Probability, Springer, vol. 20(1), pages 65-86, March.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:1:d:10.1007_s10959-006-0046-x
    DOI: 10.1007/s10959-006-0046-x
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    Cited by:

    1. Michele Baldini, 2007. "On the Lyapunov Exponent of a Multidimensional Stochastic Flow," Journal of Theoretical Probability, Springer, vol. 20(2), pages 327-337, June.

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