Welcome to the website of the
CRC TRR 388!

The collaborative research center TRR 388 "Rough Analysis, Stochastic Dynamics & Related Fields" started on the 1st of October, 2024.

This is a joint initative of TU Berlin, HU Berlin and FU Berlin.

December 18:
Winter Lectures by Alessandra Frabetti (Université Claude Bernard Lyon 1)

Location: WIAS (Erhard-Schmidt lecture room downstairs)

Talk 1 — 11:00–12:30

Transport maps on manifolds using groupoids and direct connections

In the theory of Regularity Structures, local solutions of (singular and stochastic) PDEs at different points of a base manifold are related by means of some transport maps. On non-flat manifolds, such operators are best described by Lie groupoids, a bi-fibred version of Lie groups where each element has a source and a target point on the manifold. After a short review of two

motivating examples, we shall introduce the basic mathematical tools for such a geometric approach: groupoid actions on vector bundles and direct connections on groupoids giving rise to transport maps.

Talk 2 — 2:00–2:45 pm & 3:00–3:45 pm

(with a short break for coffee, cookies, and conversation)

Direct connections on jet groupoids and regularity structures

Solutions of PDEs obtained by means of some local Taylor expansion are geometrically naturally described in terms of jet bundles. In order to relate such local solutions, we consider direct connections on jet groupoids. In this talk we shall present a natural jet prolongation of direct connections to jet groupoids. We shall also provide an example of a direct connection on a jet groupoid which does not arise in the context of Regularity Structures.

Zoom Link: https://wias-berlin-de.zoom-x.de/j/67176078712?pwd=uDar4nhdxKdS8i9Lv3h73gwA0qcOTF.1

Stochastic dynamics builds on probability theory and Itô’s stochastic analysis to study the evolution of systems under the influence of randomness, and has flourished for decades with profound impact on many fields, including statistical physics, mathematical finance, uncertainty quantification, quantum field theory, mathematical biology, economics.

Rough analysis, on the other hand, stands for recent breakthroughs in mathematics, rooted in Lyons’ rough path theory. With the original motivation of introducing robustness in noise/signal, rough analysis offers a nonlinear extension of distribution theory that is crucial for understanding singular stochastic dynamics and their possible renormalizations, and to capture nonlinear effects of signals. Transcending its origins, rough analysis recently saw the emergence of deep mathematical structures with significant geometric and algebraic components.

Together, they form the fertile grounds for this TRR Rough Analysis, Stochastic Dynamics and Related Fields.

With an intense interplay of analysis, algebra/geometry and probability theory together with closely related applied topics, such as statistics, robust modeling under uncertainty, and stochastic control theory and mathematical finance, the overarching goal of this TRR is to foster mutually beneficial interactions with the new field of rough analysis.