# Quenched invariance principle for symmetric diffusions in a degenerate random environment

The purpose of this series of lectures is to present some recent results on almost sure invariance principle for a diffusion in divergence form with measurable unbounded and degenerate ergodic random coefficients. The method combines arguments both probabilistic, such as martingale convergence theorems, and analytical, such as the Sobolev inequalities and the Moser iteration. A further result dealing with the asymptotic of the 2-d Ginzburg-Landau model will also be discussed.

Lecture 1. The homogenization problem of diffusion in divergence forms,

and the construction of the process using Dirichlet forms.

Lecture 2. The introduction of harmonic coordinates and corrector, martingale invariance principle and the ergodicity of the environment process viewed from the diffusion.

Lecture 3. The sublinear control of the corrector via Moser iteration technique.

Lecture 4. The local limit theorem and the key analytical tool of the parabolic Harnack inequality.

Lecture 5. Asymptotic of the covariance of the 2-d Ginzburg-Landau model using the random walk representation and lthe ocal limit theorem.

**References: with A. Chiarini**

Invariance Principle for symmetric Diffusions in a degenerate and

unbounded stationary and ergodic Random Medium.

http://arxiv.org/abs/1410.4483

Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium.

http://arxiv.org/abs/1501.03476

Phd Thesis at TU Berlin

https://opus4.kobv.de/opus4-tuberlin/frontdoor/index/index/docId/7219

Talk at TU Munich (slides)

http://page.math.tu-berlin.de/~chiarini/Seminars/SeminarTUM2015.pdf

Thesis defence (slides)

http://page.math.tu-berlin.de/~chiarini/Seminars/Defence15092015.pdf

Quenched Local CLT in EJP

http://ejp.ejpecp.org/article/view/4190

Quenched Invariance Principle in AIHP

http://arxiv.org/abs/1410.4483