T-ERC_STG - Tremplin-ERC Starting

Low regularity dynamics via decorated trees – LoRDeT

Submission summary

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.

Project coordination

Yvain Bruned (Institut Elie Cartan de Lorraine)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


IECL Institut Elie Cartan de Lorraine

Help of the ANR 33,103 euros
Beginning and duration of the scientific project: February 2023 - 24 Months

Useful links

Explorez notre base de projets financés



ANR makes available its datasets on funded projects, click here to find more.

Sign up for the latest news:
Subscribe to our newsletter