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

Abstract : 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.
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SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2015, 47 (3), pp.2022-2043. 〈10.1137/130949683〉
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Soumis le : lundi 8 février 2016 - 16:39:10
Dernière modification le : lundi 24 septembre 2018 - 12:26:01

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Moustapha Ba, Pierre Mathieu. A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2015, 47 (3), pp.2022-2043. 〈10.1137/130949683〉. 〈hal-01270967〉

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