The Allen-Cahn equation with random critical initial datum

  • Simon Gabriel (University of Warwick)
E1 05 (Leibniz-Saal)


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.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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