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dc.contributor.authorKowalczyk, Michal
dc.contributor.authorDolbeault, Jean
HAL ID: 87
ORCID: 0000-0003-4234-2298
dc.contributor.authorBlanchet, Adrien
dc.subjectlogarithmic Sobolev inequalities
dc.subjectHolley-Stroock perturbation results
dc.subjectspectral gap
dc.subjectPoincaré inequality
dc.subjectsharp constants
dc.subjectfunctional inequalities
dc.subjectintermediate asymptotics
dc.subjecteffective diffusion
dc.subjecttraveling front
dc.subjecttraveling potential
dc.subjectdoubly-periodic equation
dc.subjectasymptotic expansion
dc.subjectmoment estimates
dc.subjectFokker-Planck equation
dc.subjectmolecular motors
dc.subjectBrownian ratchets
dc.subjecttwo-scale convergence
dc.subjectminimizing sequences
dc.subjectdefect of convergence
dc.subjectloss of compactness
dc.subjectStochastic Stokes' driften
dc.titleStochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchetsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherUniversité des Sciences et de Technologies de Lille - Lille 1;France
dc.contributor.editoruniversityotherUniversidad de Chile;Chili
dc.description.abstractenA periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization 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 towards 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 behaviour 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 the 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.en
dc.relation.isversionofjnlnameSIAM Journal on Mathematical Analysis
dc.subject.ddclabelProbabilités et mathématiques appliquéesen

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