COLLOQUIUM
3:00 p.m., Friday (November 9, 2007)
Math Annex 1100
Marek Biskup
UCLA and Microsoft
Parabolic Anderson Model: eigenvalue order statistics and path localization
Abstract:
The Parabolic Anderson Model is a class of diffusion problems driven by
the Anderson Hamiltonian \Delta+\xi, where \Delta is the Laplacian
and \xi a random field. I will give an overview of recent results on this
model in the case when the underlying spatial structure is the hypercubic
lattice \mathbb Z^d and the \xi's are i.i.d. random variables. Then I will
report on a recent progress in the understanding of the random walk associated,
via the FeynmanKac formula, to the diffusion equation \partial_t u=\Delta u+\xi u.
Here Sznitman's technique of enlargement of obstacles generally shows
that the walk up to time t localizes in one of t^{o(1)} localization
centra within a ball of radius t^{1+o(1)}. In a specific class of potentials
(doubleexponential upper tail) I will show that there is typically only one
such localization center and will characterize its distribution. The key step
is the description of extreme order statistics of the eigenvalues of \Delta+\xi
restricted to a large finite box. Joint work with W.K\"onig.
Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).
