A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2015

A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

Résumé

We consider a diffusion process in $\mathbb{R}^d$ with a generator of the form $ L:=\frac 12 e^{V(x)}div(e^{-V(x)}\nabla ) $ where $V$ is measurable and periodic. We only assume that $e^V$ and $e^{-V}$ are locally integrable. We then show that, after proper rescaling, the law of the diffusion converges to a Brownian motion for Lebesgue almost all starting points. This pointwise invariance principle was previously known under uniform ellipticity conditions (when $V$ is bounded), and was recently proved under more restrictive $L^p$ conditions on $e^V$ and $e^{-V}$. Our approach uses Dirichlet form theory to define the process, martingales and time changes and the construction of a corrector. Our main technical tool to show the sub-linear growth of the corrector is a new weighted Sobolev type inequality for integrable potentials. We heavily rely on harmonic analysis technics.

Dates et versions

hal-01270967 , version 1 (08-02-2016)

Identifiants

Citer

Moustapha Ba, Pierre Mathieu. A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential. SIAM Journal on Mathematical Analysis, 2015, 47 (3), pp.2022-2043. ⟨10.1137/130949683⟩. ⟨hal-01270967⟩
112 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More