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

Handle URI:
http://hdl.handle.net/10754/599738
Title:
Stochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets
Authors:
Blanchet, Adrien; Dolbeault, Jean; Kowalczyk, MichaŁ
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.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Mathematical Analysis
Issue Date:
Jan-2009
DOI:
10.1137/080720322
Type:
Article
ISSN:
0036-1410; 1095-7154
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.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorBlanchet, Adrienen
dc.contributor.authorDolbeault, Jeanen
dc.contributor.authorKowalczyk, MichaŁen
dc.date.accessioned2016-02-28T06:08:41Zen
dc.date.available2016-02-28T06:08:41Zen
dc.date.issued2009-01en
dc.identifier.citationBlanchet 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.en
dc.identifier.issn0036-1410en
dc.identifier.issn1095-7154en
dc.identifier.doi10.1137/080720322en
dc.identifier.urihttp://hdl.handle.net/10754/599738en
dc.description.abstractA 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.en
dc.description.sponsorshipThis 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.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectAsymptotic expansionen
dc.subjectBrownian ratchetsen
dc.subjectContractionen
dc.subjectDoubly periodic equationen
dc.subjectEffective diffusionen
dc.subjectFokker-planck equationen
dc.subjectIntermediate asymptoticsen
dc.subjectMolecular motorsen
dc.subjectMoment estimatesen
dc.subjectStochastic stokes' driften
dc.subjectTransporten
dc.subjectTraveling fronten
dc.subjectTraveling potentialen
dc.titleStochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchetsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Mathematical Analysisen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionCentre de Recherche en Mathematiques de la Decision, Paris, Franceen
dc.contributor.institutionUniversidad de Chile, Santiago, Chileen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.