Abstract: We consider the Allen-Cahn equation with white noise initial datum in a weak coupling regime. 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/series has a positive contribution to the solution.
In this talk, we present an approach that keeps track of each summand’s contribution, using the notion of rooted trees, and determine their non-trivial Gaussian fluctuations exactly. Furthermore, by exploiting the structure of the equation, we approximate the infinite series while controlling the imposed error, and determine its limiting law.
The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.