IS27: Rough Analysis

date: 7/17/2025, time: 16:00-17:30, room: IM A

Organizer: Peter Friz (TU and WIAS Berlin)

Chair: Carlo Bellingeri (University of Lorraine)

Overcoming the order barrier for SPDEs with additive space-time white noise

Helena Katharina Kremp (TU and WIAS Berlin)

We consider strong approximations of 1 + 1-dimensional stochastic PDEs driven by additive space-time white noise. It has long been proposed \([1, 2]\), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. We prove that for a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) 1 can be achieved, a major improvement on the rate 1/4 that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) 1/2 as it is standard in the literature. The talk is based on the preprint \([3]\).
Bibliography

\([1]\) A. M. Davie and J. G. Gaines. "Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations." Mathematics of Computation, 70, no. 233, (2001), 121–135.

\([2]\) A. Jentzen and P. E. Kloeden. "Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465, no. 2102, (2008), 649–667.

\([3]\) A. Djurdjevac, H. Kremp and M. Gerencsér. "Higher order approximation of nonlinear SPDEs with additive space-time white noise" arXiv preprint arXiv:2406.03058v2.

Nonlinear rough Fokker–Planck equations

Fabio Bugini (TU Berlin)

McKean–Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker–Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and conditional law dependence, the evolution becomes random and is governed by a stochastic Fokker–Planck equation; that is, a nonlinear, nonlocal SPDE in the space of measures.

Well-posedness of such SPDEs is a difficult problem, the best result to date due to Coghi-Gess (2019), which however comes with dimension-dependent regularity assumptions. In our work, we show how rough path techniques can circumvent these entirely. Hence, and somewhat contrarily to common believe, the use of rough paths leads to substantially less regularity demands on the coefficients than methods rooted in classical stochastic analysis methods.

This is joint work with Peter K. Friz and Wilhelm Stannat.

New algebraic structures in rough analysis and their applications

Carlo Bellingeri (Université de Lorraine)

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.