Title :

Stochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchets

Author(s) :

Blanchet, Adrien [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Dolbeault, Jean [Auteur]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Kowalczyk, Michal [Auteur]

Journal title :

SIAM Journal on Mathematical Analysis

Pages :

46-76

Publisher :

Society for Industrial and Applied Mathematics

Publication date :

2009

ISSN :

0036-1410

English keyword(s) :

loss of compactness
Stochastic Stokes' drift
Brownian ratchets
molecular motors
asymptotic expansion
doubly-periodic equation
Fokker-Planck equation
moment estimates
contraction
transport
traveling potential
traveling front
effective diffusion
intermediate asymptotics
functional inequalities
sharp constants
Poincaré inequality
spectral gap
generalized Poincaré inequalities
Holley-Stroock perturbation results
logarithmic Sobolev inequalities
interpolation
perturbation
homogenization
two-scale convergence
minimizing sequences
defect of convergence
loss of compactness.

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

A 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. ...
Show more >A 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.Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL