Lectures

Seminar information archive ~03/03Next seminarFuture seminars 03/04~


2012/06/17

09:45-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
James Nolen (Duke University) 09:45-17:30
Fluctuation of solutions to PDEs with random coefficients (Part 1) (JAPANESE)
[ Abstract ]
For PDEs with random coefficients, it is interesting to understand whether the solutions exhibit some universal statistical behavior that is independent of the details of the coefficients. In particular, how do solutions fluctuate around the mean behavior? We will discuss this issue in the context of three examples:

(1) Traveling fronts in random media in one dimension.
(2) Elliptic homogenization problems.
(3) Random Hamilton-Jacobi equations.

The relation between PDE tools and probabilistic ideas will be
explained.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Leonid Ryzhik (Stanford Univeristy) 13:00-14:45
Weak coupling limits for particles and PDEs (Part 1) (JAPANESE)
[ Abstract ]
Weak random fluctuations in medium parameters may lead to a non-trivial effect after large times and propagation over long distances. We will consider several examples when the large time limit can be treated:

(1) a particle in a weakly random velocity field.
(2) weak random fluctuations of Hamilton equations, and
(3) the linear Scrhoedinger equation with a weak random potential.

The role of long range correlation of the random fluctuations will also be discussed.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/
Gregoire Nadin (CNRS / Paris 6) 15:15-17:15
Asymptotic spreading for heterogeneous Fisher-KPP reaction-diffusion equations (JAPANESE)
[ Abstract ]
The solutions of the heterogeneous Fisher-KPP equation associated with compactly supported initial data are known to take off from the unstable steady state 0 and to converge to the steady state 1 for large times. The aim of this lecture is to estimate the speed at which the interface between 0 and 1 spreads.

Using the new notion of generalized principal eigenvalues for non-compact elliptic operators, we will derive such estimates which will be proved to be optimal for several classes of heterogeneity such as periodic, almost periodic or random stationary ergodic ones.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/