# Mathematics Colloquia and Seminars

### Long-time tails for diffusions in random media: parabolic

Mathematical Physics & Probability

 Speaker: Marek Biskup, Microsoft Research Location: 693 Kerr Start time: Mon, Jan 15 2001, 4:10PM

I will discuss the long-time asymptotics (both of the moments and almost sure) of the solution to a parabolic second-order differential problem (so called parabolic Anderson model) on $\mathbb Z^d$ with a random i.i.d. potential, bounded from above. Both asymptotics are determined by appropriate variational principles. As an application, the Lifshitz tails for the spectrum of the associated random Schroedinger operator (Anderson Hamiltonian) can explicitly be computed. The results extend various findings about the "simple random walk among Poissonian obstacles" obtained by Sznitman and his school in the 1990s.