Random Walks and Homogenization Theory

Description:

The central-limit type results are universal in many random walk models: they are known as Donsker’s theorem for the classical Zd random walk, and also hold for some random walks in random environment, even when the environment is degenerate like percolation. In their proofs, the homogenization theory, which comes from PDE, plays an important role. This course will cover the following topics:

1. Donsker’s theorem of classical random walk (2 lectures)

2. Some examples of random walks in random environments (2 lectures)

3. Homogenization theory (5 lectures)

4. Random walk on percolation cluster (3 lectures)

Prerequisite：

Basic probability and PDE

Reference：

1. Armstrong, S., Kuusi, T. and Mourrat, J.C., 2019. Quantitative stochastic homogenization and large-scale regularity (Vol. 352). Springer.

2. Biskup, M., 2011. Recent progress on the random conductance model. Probability Surveys, 8, pp.294-373.

3. Dario, P. and Gu, C., 2021. Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters. The Annals of Probability, 49(2), pp.556-636.

4. Jikov, V.V., Kozlov, S.M. and Oleinik, O.A., 2012. Homogenization of differential operators and integral functionals. Springer Science & Business Media.

5. Lawler, G.F. and Limic, V., 2010. Random walk: a modern introduction (Vol. 123). Cambridge University Press.