  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.
