Stochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchets
Kowalczyk, Michal; Dolbeault, Jean; Blanchet, Adrien (2009), Stochastic Stokes' drift, homogenized functional inequalities, and large time behavior of Brownian ratchets, SIAM Journal on Mathematical Analysis, 41, 1, p. 46-76. http://dx.doi.org/10.1137/080720322
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00270521/en/
Journal nameSIAM Journal on Mathematical Analysis
MetadataShow full item record
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. 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.
Subjects / Keywordsperturbation; interpolation; logarithmic Sobolev inequalities; Holley-Stroock perturbation results; spectral gap; Poincaré inequality; sharp constants; functional inequalities; intermediate asymptotics; effective diffusion; transport; contraction; traveling front; traveling potential; doubly-periodic equation; asymptotic expansion; moment estimates; Fokker-Planck equation; molecular motors; Brownian ratchets; homogenization; two-scale convergence; minimizing sequences; defect of convergence; loss of compactness; Stochastic Stokes' drift
Showing items related by title and author.
A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients Kowalczyk, Michal; Illner, Reinhard; Dolbeault, Jean; Bartier, Jean-Philippe (2007) Article accepté pour publication ou publié