Rough Invariant Manifolds for Stochastic Kuramoto-type Models

• Applicants: Professorin Dr. Alexandra Blessing; Professor Christian Kühn, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 543163250

Invariant manifolds are key geometric objects to understand the phase space of dynamical systems. It is technically very challenging to transfer deterministic invariant manifold methods to the setting of stochastic systems. Yet, recent progress has shown that there are possible approaches, e.g., results by the two PIs on combining rough paths with random dynamical systems ideas provide one possible bridge. In this project, we are going to study stochastic invariant manifolds for stochastic Kuramoto-type models, which are particle systems arising across all sciences having their origins in non-equilibrium statistical physics. For example, Kuramoto-type models naturally arise in the process of phase reduction for coupled oscillators. In particular, we are going to investigate stable, unstable and center stochastic/rough manifolds in various settings. Starting from analyzing spectral gap conditions and parametric dependence, we proceed to consider the impact of delay-coupling and non-Markovian noise structures. To provide efficient practical manifold representations, we are going to develop analytical approximation techniques via series and forward-backward schemes. Finally, we are going to also explore first steps towards (singular) limits from finite- to infinite-dimensional stochastic systems for stochastic Kuramoto-type models including the infinite-particle mean-field limit, time-scale separation limits, more complex coupling topologies and for degenerate noise levels. In summary, our project aims to provide more technical tools for, and a much more coherent view of the stochastic phase space geometry induced by particle systems in statistical physics.

• DFG Programme: Research Grants