%0 Journal Article
%T A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential
%+ Institut de Mathématiques de Marseille (I2M)
%A Ba, Moustapha
%A Mathieu, Pierre
%Z Publication issue de la thèse de Moustapha Ba
%< avec comité de lecture
%@ 0036-1410
%J SIAM Journal on Mathematical Analysis
%I Society for Industrial and Applied Mathematics
%V 47
%N 3
%P 2022-2043
%8 2015
%D 2015
%Z 1312.4817
%R 10.1137/130949683
%K Sobolev inequality
%K invariance principle
%K diffusions
%K homogenization
%K periodic potential
%Z Mathematics [math]/Probability [math.PR]
%Z Mathematics [math]/Analysis of PDEs [math.AP]Journal articles
%X 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.
%G English
%L hal-01270967
%U https://hal.science/hal-01270967
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ TDS-MACS