PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Leonid Berlyand.

Title:A quantitative theory of stochastic homogenization
Seminar:Marker Lecture Series
Speaker:Dr. Felix Otto, Max-Planck-Institute for Mathematics
If one is interested in the conductivity of a composite, say, one has to solve an elliptic equation with coefficients that vary on a length scale much smaller than the characteristic scale of the domain. We are interested in a situation where the coefficients are characterized in statistical terms: Their statistics are assumed to be translation invariant and to decorrelate over large distances. As is known by qualitative theory, the solution operator behaves -- on large scales -- like the solution operator of an elliptic problem with homogeneous, deterministic coefficients, a huge reduction in complexity! Theory provides a formula for the homogenized coefficients, based on the construction of a "corrector", which defines harmonic coordinates. In practise, this formula has to be approximated by a "Representative Volume Element", leading to a random and a systematic error. We present optimal estimates of both. We are also interested in other quantitative aspects: How close is the actual solution to the homogenized one --- we give an optimal answer, and point out the connections with elliptic regularity theory (input from Nash's theory, a new outlook on De Giorgi's theory). We are also interested in the quantitative ergodicity properties for the process usually called "the environment as seen from the random walker". We give an optimal estimate that relies on a link with (the Spectral Gap for) another stochastic process on the coefficient fields, namely heat-bath Glauber dynamics. This connection between statistical mechanics and stochastic homogenization has previously been used in opposite direction (i. e. with qualitative stochastic homogenization as an input). This is joint work with A. Gloria, S. Neukamm, and D. Marahrens.

Room Reservation Information

Room Number:MB114
Date:10 / 22 / 2013
Time:03:35pm - 04:35pm