Algebraic deformation for (S)PDEs

Abstract

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in Regularity Structures in the work arXiv:1610.08468 and in arXiv:2005.01649 for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure in arXiv:1610.08468 can be viewed as a deformation of the Butcher-Connes-Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin-Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.

To appear in the Journal of the Mathematical Society of Japan

References in corpus (9)

• Langevin dynamic for the 2D Yang-Mills measure
• Resonance based schemes for dispersive equations via decorated trees
• The structure group for quasi-linear equations via universal enveloping algebras
• A priori bounds for the ${\Phi }^4$ equation in the full sub-critical regime
• Renormalisation from non-geometric to geometric rough paths
• Renormalised singular stochastic PDEs
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• Locality for singular stochastic PDEs
• Bogoliubov type recursions for renormalisation in regularity structures

Cited by in corpus (10)

• The structure group for quasi-linear equations via universal enveloping algebras
• A Survey on the Munthe-Kaas-Wright Hopf Algebra
• Renormalised singular stochastic PDEs
• Multi-indice B-series
• Locality for singular stochastic PDEs
• Approximations of dispersive PDEs in the presence of low-regularity randomness
• Parametrization of renormalized models for singular stochastic PDEs
• Multi-indices coproducts from ODEs to singular SPDEs
• Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
• Flows driven by multi-indices Rough Paths