The Allen-Cahn equation with random critical initial datum

We consider the Allen-Cahn equation with white noise initial datum under critical rescaling. The usual approach of performing a Picard iteration of the solution yields an infinite series of stochastic iterated integrals. In contrast to considering initial datum under sub-critical rescaling, each term in the infinite expansion has a positive contribution to the solution. In this talk we will give a gentle introduction into the methods used to control such an infinite series. We start by identifying stochastic integrals with rooted trees. We will then treat some explicit examples that motivate a systematic approach of determining the statistics of each term in the expansion. Time permitting, we identify the distribution of the solution with the help of a series expansion of an explicit ODE, exploiting the structure of the initial PDE.

The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.