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Séminaire du LJLL - 01 04 2022 14h00 : F. Otto

Vendredi 01 avril 2022 — 14h00
Exposé donné en personne dans la salle du séminaire du Laboratoire Jacques-Louis Lions
avec diffusion simultanée par Zoom

Felix Otto (Institut Max Planck pour les mathématiques en sciences, Leipzig)
Regularity structures without Feynman diagrams
Résumé
Singular stochastic PDEs are those stochastic PDEs in which the noise is so rough that the nonlinearity requires a renormalization. Hairer’s regularity structures provide a framework for the solution theory. His notion of a model can be understood as providing a (formal) parametrization of the entire solution manifold of the renormalized equation. In this talk, I will focus on the stochastic estimates of the model.
I shall present a more analytic than combinatorial approach : Instead of using trees to index the model, we consider all partial derivatives with respect to the function defining the nonlinearity (and thus work with multi-indices as index set). Instead of a Gaussian calculus guided by Feynman diagrams arising from the trees, we consider first-order partial derivatives with respect to the noise, i.e. Malliavin derivatives.
We employ tools from quantitative stochastic homogenization like spectral gap estimates, which naturally complement the standard choice of renormalization, and annealed estimates, which as opposed to their quenched counterparts preserve scaling.
This is joint work with P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.