## Seminars and Colloquia by Series

Thursday, April 18, 2019 - 15:05 , Location: Skiles 006 , Nizar Demni , University of Marseille , Organizer: Christian Houdre
Thursday, April 11, 2019 - 15:05 , Location: Skiles 006 , , Northeastern University , , Organizer: Michael Damron
Thursday, April 4, 2019 - 15:05 , Location: Skiles 006 , Mayya Zhilova , School of Mathematics, GaTech , Organizer: Christian Houdre
XXX
Thursday, March 28, 2019 - 15:05 , Location: Skiles 006 , Liza Rebova , Mathematics, UCLA , Organizer: Christian Houdre
Thursday, March 14, 2019 - 15:05 , Location: Skiles 006 , Galyna Livshyts , SOM, GaTech , Organizer: Christian Houdre
Thursday, March 7, 2019 - 15:05 , Location: Skiles 006 , Samy Tindel , Purdue University , Organizer: Christian Houdre
In this talk I will first recall some general facts about the parabolic Anderson model (PAM), which can be briefly described as a simple heat equation in a random environment. The key phenomenon which has to be observed in this context is called localization. I will review some ways to express this phenomenon, and then single out the so called eigenvectors localization for the Anderson operator. This particular instance of localization motivates our study of large time asymptotics for the stochastic heat equation. In the second part of the talk I will describe the Gaussian environment we consider, which is rougher than white noise, then I will give an account on the asymptotic exponents we obtain as time goes to infinity. If time allows it, I will also give some elements of proof.
Thursday, February 28, 2019 - 15:05 , Location: Skiles 006 , , Indiana University, Bloomington , , Organizer: Michael Damron
Thursday, February 21, 2019 - 15:05 , Location: Skiles 006 , , University of Rochester , , Organizer: Michael Damron
Thursday, February 14, 2019 - 15:05 , Location: Skiles 006 , C. Houdre , SOM, GaTech , Organizer: Christian Houdre
Thursday, February 7, 2019 - 15:05 , Location: Skiles 006 , , Temple University , , Organizer: Michael Damron
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the&nbsp;homogenization&nbsp;of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.