Mass asymptotics for the 2d parabolic Anderson model with space white noise potential

Abstract: We study the long time behavior of the total mass of the 2d parabolic Anderson model (PAM) with white noise potential, which is the universal scaling limit of 2d branching random walks in small random environments. There are several known results on the long time behavior of the PAM for more regular potentials, but the 2d white noise is very singular and it requires renormalization techniques. In particular, the Feynman-Kac representation, usually the main tool for deriving asymptotics, breaks down. To overcome this problem we use a measure transform and we introduce a new "partial Feynman-Kac representation“. The new representation is based on a diffusion with distributional drift, and we derive Gaussian heat kernel bounds for such diffusions. Based on joint works with Wolfgang König and Willem van Zuijlen.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

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