TY - JOUR

T1 - Stochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets

AU - Blanchet, Adrien

AU - Dolbeault, Jean

AU - Kowalczyk, MichaŁ

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2009/1

Y1 - 2009/1

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/599738

UR - http://epubs.siam.org/doi/10.1137/080720322

UR - http://www.scopus.com/inward/record.url?scp=70450182951&partnerID=8YFLogxK

U2 - 10.1137/080720322

DO - 10.1137/080720322

M3 - Article

VL - 41

SP - 46

EP - 76

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 1

ER -