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 to perform the crucial renormalisation procedures. Decorated trees are used for expanding solutions of these dynamics. They also appear for describing resonance schemes for a large class of dispersive PDEs at low regularity. The aim of this project is to push forward 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:

1. Give a notion of existence and uniqueness of two classes of singular SPDEs: the quasilinear and the dispersive SPDEs.

2. Identify the process whose dynamic has the Brownian loop measure as invariant measure via an extension of the resolution of SPDEs to discrete dynamics.

3. Develop the algebraic structures for singular SPDEs in connection with Numerical Analysis, Perturbative Quantum Field Theory and Rough Paths.

4. Use decorated trees for dispersive PDEs with random initial data and provide a systematic way to derive wave kinetic equations in Wave Turbulence.

5. Develop a software platform for decorated trees and their Hopf algebraic structures that appear in singular SPDEs and dispersive PDEs.

Team Members

Photo from left to right: Yingtong Hou, Yvain Bruned, Jacob Armstrong-Goodall, Usama Nadeem.

Decorated Tree-like structures for singular dynamics: 27 to 29 of May 2024

Conference organised in Nancy funded by the ERC LoRDeT. Programme available at pdf.


Below a past workshop which has been funded by the ANR and it is connected to the themes of the ERC LoRDeT:

Hopf algebras, operads, deformations for singular dynamics: 21 to 23 of June 2023

Conference organised in Nancy funded by the ANR via the project LoRDeT (Dynamiques de faible regularité via les arbres décorés) from the projects call T-ERC_STG. PI of the project: Yvain Bruned. Programme available at pdf.