IS27: Rough Analysis

Organizer: Peter Friz (TU and WIAS Berlin)

New algebraic structures in rough analysis and their applications

Carlo Bellingeri

Rough analysis has deeply changed the study of differential equations driven by highly irregular signals, such as sample paths of stochastic processes. At the heart of this theory lies a rich algebraic framework that encodes the nonlinear interactions of the driving signals and enables robust solution theories for SDEs and SPDEs. In recent years, new algebraic structures have emerged as powerful tools for describing general compositions of these solutions with nonlinear functions. In this talk, I will provide an overview of these emerging structures, explain how they naturally arise in the study of singular stochastic dynamics, and illustrate their applications in the contexts of branched rough paths and the generalized KPZ equation, as developed in articles \([1,2]\).

Bibliography

\([1]\) Carlo Bellingeri, Yvain Bruned. “Symmetries for the gKPZ equation via multi-indices." arXiv preprint arXiv:2410.00834, 2024, pp. 32.

\([2]\) Carlo Bellingeri, Emilio Ferrucci, Nikolas Tapia. “Branched Itô formula and natural Itô-Stratonovich isomorphism." arXiv preprint arXiv:2312.04523, 2023, pp. 54.

Strong regularization of differential equations with integrable drifts by fractional noise

Le

We consider stochastic differential equations with integrable time-dependent drift driven by additive fractional Brownian noise whose Hurst parameter is less than 1/2. Under some subcriticality conditions, it is shown that such equation has a unique pathwise solution. Furthermore, stability with respect to all parameters is established. Our strong uniqueness result can be considered as an extension of that from Krylov and Röckner (2005) for Brownian motion, it also improves upon previous results of Nualart and Ouknine (2003) for dimension one. Our methods are built upon Lyons’ rough path theory and the stochastic sewing lemma, complemented by the quantitative John–Nirenberg inequality for stochastic processes of vanishing mean oscillation. Joint work with Oleg Butskovsky and Toyomu Matsuda.