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    Stochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets

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    Type
    Article
    Authors
    Blanchet, Adrien
    Dolbeault, Jean
    Kowalczyk, MichaŁ
    Date
    2009-01
    Permanent link to this record
    http://hdl.handle.net/10754/599738
    
    Metadata
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    Abstract
    A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincarae and logarithmic Sobolev inequalities in the homogeniz ation limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates toward s equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes' drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behavior of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in self-similar variables attached to t he center of mass to a stationary solution of a Fokker-Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes' drift with given potential flow. © 2009 Society for Industrial and Applied Mathematics.
    Citation
    Blanchet A, Dolbeault J, Kowalczyk M (2009) Stochastic Stokes’ Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets. SIAM J Math Anal 41: 46–76. Available: http://dx.doi.org/10.1137/080720322.
    Sponsors
    This author’s research was partially supported by the KAUST investigator award.This author’s research was partially supported by FONDECYT 1050311, Nucleo Milenio P04-069-F, FONDAP, and ECOSCONYCIT# C05E05.
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Mathematical Analysis
    DOI
    10.1137/080720322
    ae974a485f413a2113503eed53cd6c53
    10.1137/080720322
    Scopus Count
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