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Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

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Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization 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.