GSIS - Winter School 2017 - Sendai

Qualitative and Quantitative Stochastic Homogenization

On this site you find course material regarding a lecture series (15h) on stochastic homogenization within the GSIS International Winter School 2017 (at the Center for Pure and Applied Mathematics (RCPAM) / Graduate School of Information Sciences (GSIS), Tohoku University, Sendai, Japan).

Content of the course (tentative)

• Introduction: From micro to macro scale models
• Homogenization of elliptic equations
  • for periodic coefficients - a corrector based approach
  • for stationary and ergodic coefficients - sublinearity of the corrector
  • two-scale expansion and error representation - the extended corrector
• Quantitative stochastic homogenization (discrete setting)
  • quantifcation of ergodicity via spectral gap
  • semigroup approach

Course material

• A tentative version of the lecture notes (Version Feb 16 2017) can be found here
• The introductory presentation can be found here
• The presentation regarding the last lecture can be found here

Preliminaries

• Basic knowledge in functional analysis and linear elliptic partial differential equations; for a summary see appendix in here

Material for the tutorial

• The problem sheet can be found here
• If you would like to contribute a solution to one of the problems, please use this LaTex-template and send me the file. We might add it as a supplement to the lecture notes.

Literature on the topic

• Antoine Gloria, Stefan Neukamm and Felix Otto: Quantification of ergodicity in stochastic homogenization : optimal bounds via spectral gap on Glauber dynamics In: Inventiones mathematicae, 199 (2015) 2, p. 455-515
  long version: 3/2013
• G.C. Papanicolaou and S.R.S. Varadhan. [Pap1979]
  Boundary value problems with rapidly oscillating random coefficients.
  In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. Janos Bolyai, pages 835873. North-Holland, Amsterdam, 1981.
• François Murat and Luc Tartar:  H-convergence. Topics in the Mathematical Modelling of Composite Materials Volume 31 of the series "Progress in Nonlinear Differential Equations and Their Applications", pp 21-43