FluCo
Fluctuations in continuum and conservative stochastic partial differential equations
Fluctuations occur throughout complex systems: as thermal fluctuations in physical models, algorithmic randomness in machine learning and unresolved small-scale variability in climate dynamics. Although the microscopic details differ, many such systems exhibit common large-scale structures. FluCo seeks mathematical principles that explain this universality by deriving and analysing scaling limits that retain both the effective mean behaviour and the fluctuations around it.
The project focuses on conservative stochastic partial differential equations as universal fluctuating continuum models. These equations preserve quantities such as mass and arise naturally as mesoscopic descriptions between microscopic particle systems and deterministic macroscopic PDEs. Their analysis brings together singular SPDEs, strongly nonlinear and degenerate PDEs, kinetic formulations, supercriticality, interacting particle systems and random dynamical systems.
Scientific programme
FluCo is organized around three central challenges:
- Well-posedness: establishing robust solution theories for singular conservative SPDEs.
- Singular limits: understanding scaling limits for supercritical conservative SPDEs.
- Stochastic dynamics: developing a long-time and qualitative theory for conservative SPDEs.
These questions support a wider programme on regularity, non-equilibrium large deviations, numerical approximation and stochastic models of learning. The broader aim is to uncover structures that remain meaningful across different physical and computational systems despite their many parameters and interactions.
Project facts
| Programme | Horizon Europe — European Research Council (ERC) Main Programme |
| Call | ERC Consolidator Grants 2022 (ERC-2022-COG) |
| Grant agreement | 101088488 |
| DOI | 10.3030/101088488 |
| Period | 1 November 2023–31 October 2028 |
| Host institution | Technische Universität Berlin |
| Principal investigator | Benjamin Gess |
SAiS team
Research areas
Official sources
Related publications
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The Incompressible Navier–Stokes–Fourier System with Thermal Noise
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Large Spikes in Stochastic Gradient Descent: A Large-Deviations View
Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to split into inflationary and deflationary regimes,…
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The Porous Medium Equation: Multiscale Integrability in Large Deviations
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Probabilistically Strong Solutions to Stochastic Euler Equations
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We consider the large deviations of the hydrodynamic rescaling of the zero-range process on $\mathbb{Z}^d$ in any dimension $d\ge 1$. Under mild and canonical hypotheses on the local jump rate, we obtain matching upper and lower bounds, thus resolving the problem opened…
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Random dynamical systems for McKean–Vlasov SDEs via rough path theory
The existence of random dynamical systems for McKean–Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system…