Date/heure
16 avril 2024
16:30 - 17:30
Lieu
Salle de conférences Nancy
Oratrice ou orateur
Felix Otto (Max Planck Institute for Mathematics in the Sciences, Leipzig)
Catégorie d'évènement Colloquium
Résumé
Titre : Singular stochastic partial differential equations: more geometry and less combinatorics
Résumé :
Singular stochastic partial differential equations are those stochastic PDE in which the noise is so rough that the nonlinearity requires a renormalization. The guiding principle of renormalization is to preserve as many symmetries of the solution manifold as possible. We follow the approach of mathematical physics, and of Hairer’s regularity structures, which however we re-interpret as providing an informal parameterization of the infinite-dimensional nonlinear solution manifold.
We systematically follow this more geometric/analytic than combinatorial point-of-view: Instead of appealing to an expansion indexed by trees, we consider all partial derivatives w. r. t. the « constitutive » function defining the nonlinearity. Instead of a Gaussian calculus guided by Feynman diagrams arising from pairing nodes of two trees, we consider derivatives w. r. t. the noise, i. e. Malliavin derivatives. We interpret the Malliavin derivative of the parameterization as an approximate tangent vector to the solution manifold, which yields a sparse representation in terms of the parameterization itself, and paves the way for its stochastic estimate. Ultimately, this gives a characterization of the solution manifold that is oblivious to the divergent counter terms.
This is work with L. Broux, P. Linares, M. Tempelmayr, and P. Tsatsoulis, based on work with J. Sauer, S. Smith, and H. Weber.