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