Chenlin Gu

In the year 2022, I teach the course Random Walks and Homogenization Theory at YMSC.

Random Walks and Homogenization Theory

Some general information about this course can be found here.

Week 1 (Sept 13,14): Local central limit theorem [video] [ppt]
-general introduction, review of CLT, local central limit theorem, random walks on Zd.
Week 2 (Sept 20,21): Donsker's theorem [video]
-weak convergence in Polish space, weak convergence of stochastic process, Donsker's theorem.
Week 3 (Sept 27,28): Random walks in random environment [video]
-1d result of RWRE, random conductance model, electric network method.
Week 4 (Oct 4,5): Invariance principle for random conductance model [video]
-martingale CLT theorem, corrector method, invariance principle for random conductance model.
Week 5 (Oct 11,12): Classical results of elliptic and parabolic equation [video]
-existence of solution, variational description, regularity, heat kernel estimate.
Week 6 (Oct 18,19): Qualitative periodic homogenization [video]
-1d example, homogenized solution, corrector and flux, two-scale expansion.
Week 7 (Oct 25,26): Qualitative stochastic homogenization [video]
-ergodic theory, variational characterization of homogenized coefficient, renormalization argument.
Week 8 (Nov 1,2): Quantitative stochastic homogenization 1 [video]
-subadditive quantity and dual quantity, concentration inequality for fluctuations.
Week 9 (Nov 8,9): Quantitative stochastic homogenization 2 [video]
-multi-scale Poincare inequality, convergence of mean.
Week 10 (Nov 15,16): Introduction of percolation [video]
-phase transition, uniqueness of infinite cluster, renormalization in supercritical phase.
Week 11 (Nov 22,23): Invariance principle, local CLT for random walk on percolation cluster [video]
-argument of good cubes, Gaussian bound on percolation, convergece rate of density.
Week 12 (Nov 30, Dec 1): Further discussions [video]
-invited talks by Scott Armstrong, Xin Chen and Jian Wang.