Periodic Reporting for period 1 - LoRDeT (Low Regularity Dynamics via Decorated Trees)
Okres sprawozdawczy: 2023-09-01 do 2026-02-28
The project focuses on Low regularity dynamics such as singular stochastic partial differential equations and dispersive partial differential equations with random initial data.
The low regularity comes from a singular (random) noise or a singular (random) initial value. There are many examples where low regularity is needed to encode natural phenomena like random
growing interfaces and dynamics of the Euclidean quantum field theory, among others.
The objectives of the project are to push forward a combinatorial approach based on decorated trees. The idea is that one has to expand the solution of these singular dynamics via (stochastic) iterated integrals and perform analytical manipulations on them. One wants to abstract this procedure via the use of well-chosen Hopf algebraic structures to propagate analytical estimates from small terms to bigger ones.
Such a program has been achieved in specific cases, like the theory of Regularity Structures invented by Martin Hairer for parabolic singular SPDEs and the resonance-based schemes for dispersive PDEs with low regularity initial data. One wants to extend this approach to more equations, such as quasilinear and dispersive equations. This will propose a unified way to treat low regularity dynamics via combinatorial tools.
The low regularity comes from a singular (random) noise or a singular (random) initial value. There are many examples where low regularity is needed to encode natural phenomena like random
growing interfaces and dynamics of the Euclidean quantum field theory, among others.
The objectives of the project are to push forward a combinatorial approach based on decorated trees. The idea is that one has to expand the solution of these singular dynamics via (stochastic) iterated integrals and perform analytical manipulations on them. One wants to abstract this procedure via the use of well-chosen Hopf algebraic structures to propagate analytical estimates from small terms to bigger ones.
Such a program has been achieved in specific cases, like the theory of Regularity Structures invented by Martin Hairer for parabolic singular SPDEs and the resonance-based schemes for dispersive PDEs with low regularity initial data. One wants to extend this approach to more equations, such as quasilinear and dispersive equations. This will propose a unified way to treat low regularity dynamics via combinatorial tools.
In the project, we have developed a general theory of solutions for quasilinear parabolic SPDEs in the full sub-critical regime and identified the main symmetry one wants to impose on solutions of these equations, namely, the chain rule symmetry. This symmetry allows us to get local counter-terms in the renormalised equations. It also provides global well-posedness for the space-time white noise.
Such an achievement has been possible thanks to a full characterisation of the chain rule. This characterisation is obtained via ideas from homological algebra and operad theory. The main new insight is to describe the space of invariant terms homologically, using a suitable perturbation of the differential of the operadic twisting of some well-chosen operad.
Another main achievement is a thorough understanding of multi-indices, a new combinatorial structure that has emerged for studying scalar-valued low regularity dynamics. We have investigated the appearance of this new combinatorial set in Quantum Field Theory and Numerical Analysis. We have results saying that they may not exist other combinatorial sets lying between multi-indices and decorated trees, making these two sets unique for studying singular dynamics.
We have also made important progress in understanding discretisation of singular SPDEs, proposing a semi-general convergence theorem for the generalized KPZ discrete model in regularity structures.
Concerning dispersive equations, we have identified the main algebraic structure used for deriving normal forms, which is the arborification coming from the mould calculus introduced by Jean Ecalle.
This is an important step toward a better combinatorial understanding of dispersive PDEs with random initial data and Wave turbulence.
Such an achievement has been possible thanks to a full characterisation of the chain rule. This characterisation is obtained via ideas from homological algebra and operad theory. The main new insight is to describe the space of invariant terms homologically, using a suitable perturbation of the differential of the operadic twisting of some well-chosen operad.
Another main achievement is a thorough understanding of multi-indices, a new combinatorial structure that has emerged for studying scalar-valued low regularity dynamics. We have investigated the appearance of this new combinatorial set in Quantum Field Theory and Numerical Analysis. We have results saying that they may not exist other combinatorial sets lying between multi-indices and decorated trees, making these two sets unique for studying singular dynamics.
We have also made important progress in understanding discretisation of singular SPDEs, proposing a semi-general convergence theorem for the generalized KPZ discrete model in regularity structures.
Concerning dispersive equations, we have identified the main algebraic structure used for deriving normal forms, which is the arborification coming from the mould calculus introduced by Jean Ecalle.
This is an important step toward a better combinatorial understanding of dispersive PDEs with random initial data and Wave turbulence.
These preliminary achievements confirm and constrain the study of the combinatorial sets to multi-indices and decorated trees. They also reveal the importance of operad theory, species theory and homological algebra. Previously, the main algebraic tools were Hopf algebras; now with these new powerful tools one can solve open problems out of reach by other methods. One expects to see this approach more used in low regularity dynamics for studying their symmetries among other properties. One challenge that the project wants to address is a thorough understanding of the algebraic structures at play in dispersive PDEs that can cover recent results obtained in Wave turbulence. Also, one wants to have a more systematic way to construct discrete models for singular SPDEs.