Research Spotlights

Markus Tempelmayr — A diagram-free approach to the stochastic estimates in regularity structures

Veröffentlicht am 06.05.2022

We explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced by Otto, Sauer, Smith and Weber. More precisely, we construct and stochastically estimate the renormalized model, avoiding the use of Feynman diagrams but still in a fully automated, i.e., inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range.

We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modeled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. The approach is analytic and top-down rather than combinatorial and bottom-up. This is joint work with Pablo Linares, Felix Otto, and Pavlos Tsatsoulis.

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