* Title : Limit theorems for the one-dimensional parabolic Anderson model with white noise potential

* Abstract : 

 The parabolic Anderson model (PAM) is the heat equation with a random potential. It describes mass transport through a random field of sinks and sources. We consider the one-dimensional PAM

∂tu=∂x2u+ξu,u(0,⋅)=1,

where ξ is a spatial white noise. The solution is intermittent. It develops rare but very high peaks, which dominate its moments.

In this talk, we study how these peaks affect the spatial integral

U(t)=∫−L(t)/2L(t)/2u(t,x) dx,L(t)=exp⁡(α3t324),

where α>0. We show that U(t) satisfies a weak law of large numbers for α>1 and a central limit theorem for α>2. For α∈(0,2), a few high peaks dominate the fluctuations. In this case, U(t) converges in distribution, after explicit centering and scaling, to a totally asymmetric α-stable law. To the best of our knowledge, these are the first stable limit results for the continuous PAM.

The key step is a sharp tail estimate for the integral over a subinterval. It follows from two spectral results for the Anderson Hamiltonian −∂x2−ξ, on a growing interval. The first gives the lower-tail asymptotics of the lowest eigenvalue, including the exact prefactor. The second shows that, on this lower-tail event, the L1 norm of the corresponding eigenfunction concentrates around an explicit value.

This is joint work with Kumwoo Kim (POSTECH) and Jaeyun Yi (KIAS). The talk will be given in Korean.