On the invariant measure of a positive recurrent diffusion in R

2004/12/20 by Michele L. Baldini, Michele Baldini, Baldini, Michele L.
Computer Science · Mathematics · #60H15 #60J60 #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Dynamics and Pattern Formation #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60H15 #msc:60J60

paper · pdf · doi:10.48550/arxiv.math/0412410

18 pages

arxiv created 2004/12/20 · openalex publication_date 2004/12/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE X_t=x+∫₀^tσ(X_s)\strat db(s)+∫₀^t m(X_s) ds, we show that the associated stochastic flow of diffeomorphisms focuses as fast as exp(-2t∫_R(m²)/(σ²) 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.

Citations

• Random dynamical systems
• The Parabolic Differential Equations and the Associated Semi-Groups of Transformations

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