Low Regularity Dynamics via Decorated Trees

Low regularity dynamics are used for describing various physical and biological phenomena near criticality. The low regularity comes from singular (random) noise or singular (random) initial value.

The first example is Stochastic Partial Differential Equations (SPDEs) used for describing random growing interfaces (KPZ equation) and the dynamic of the euclidean quantum field theory (stochastic quantization). The second concerns dispersive PDEs with random initial data which can be used for understanding wave turbulence.

A recent breakthrough is the resolution of a large class of singular SPDEs through the theory of Regularity Structures invented by Martin Hairer.

Such resolution has been possible thanks to the help of decorated trees and their Hopf algebras structures for organising different renormalisation procedures. Decorated trees are used for expanding solutions of these dynamics.

The aim of this project is to enlarge the scope of resolution given by decorated trees and their Hopf algebraic structures. One of the main ideas is to develop algebraic tools by the mean of algebraic deformations.

We want to see the Hopf algebras used for SPDEs as deformation of those used in various fields such as numerical analysis and perturbative quantum field theory.

This is crucial to work in interaction with these various fields in order to get the best result for singular SPDEs and dispersive PDEs.

We will focus on the following long-term objectives:- Give a notion of existence and uniqueness of quasilinear and dispersive SPDEs. - Derive a general framework for discrete singular SPDEs.- Develop algebraic structures for singular SPDEs in connection with numerical analysis, perturbative quantum field theory and rough paths.- Use decorated trees for dispersive PDEs with random initial data and derive systematically wave kinetic equations in Wave Turbulence. - Develop a software platform for decorated trees and their Hopf algebraic structures.