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

Abstract

In this paper, 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 assume only 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); see [A. Bensoussan, J.-L. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North--Holland, Amsterdam, New York, 1978] and [A. Lejay, Asymptot. Anal., 28 (2001), pp. 151--162]. Our approach uses Dirichlet form theory to define the process, martingales, time changes, and construction of a corrector. Our main technical tool to show the sublinear growth of the corrector is a new weighted Sobolev-type inequality for integrable potentials. We rely heavily on harmonic analysis techniques.