Modul: MAT971 Stochastische Prozesse

## A factorization formula for the partition function in the parabolic Anderson model

Dr. Tobias Hurth talk

**Date:** 16.10.19 **Time:** 17.15 - 18.15 **Room:** ETH HG G 43

We consider a continuous-time simple symmetric random walk on the integer lattice Z^d in dimension greater than or equal to 3. The random walk is subject to a random potential induced by independent two-sided Brownian motions linked with the sites in Z^d. In the high-temperature regime, for the partition function that corresponds to starting from site x at time s and ending on site y at time t, we will state and motivate a factorization formula of the type obtained for example by Sinai (1995) and Kifer (1997) for different polymer models. We shall explain that the error term in the formula is uniformly small not just in the diffusive regime |x-y| < (t-s)^{1/2}, but up to |x - y| < (t-s)^{1-epsilon} for epsilon arbitrarily small. We will then outline how the factorization formula can be used to show that the global stationary solution to the semidiscrete stochastic heat equation attracts solutions whose initial data grows subexponentially. The talk is based on a project with Kostya Khanin and Beatriz Navarro Lameda. Fedor Nazarov helped us with the proof of the factorization formula.