CoScaRa project
Rough and nonlinear transport in stochastic fluid dynamics
Transport by irregular or random velocity fields is a central mechanism in fluid dynamics. It can enhance mixing and dissipation, alter long-time behaviour and change the effective dynamics of nonlinear systems. This project develops a unified mathematical approach to two complementary forms of stochastic transport.
Scientific programme
- Rough stochastic transport: long-time behaviour and mixing of passive scalars driven by rough random velocity fields, with particular emphasis on the Kraichnan model. A pathwise random-dynamical-systems theory is developed for SDEs with irregular coefficients.
- Nonlinear stochastic transport: nonlinear transport equations driven by random velocity fields, their derivation from microscopic particle descriptions, and extensions of random-dynamical-systems methods to McKean–Vlasov SDEs and nonlinear Fokker–Planck equations.
Together, the two strands connect rough analysis, stochastic dynamics, mean-field limits and fluid mechanics. They address how randomness affects transport across scales and how transport noise can regularize, stabilize or mix nonlinear systems.
Project facts
| Programme | DFG Priority Programmes |
| Parent programme | SPP 2410 Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness (CoScaRa) |
| DFG project number | 526086093 |
| Period | Since 2023 |
| Applicants | Benjamin Gess and Rishabh Gvalani |
| International context | Switzerland and United Kingdom |
SAiS team
Research areas
Official sources
Related publications
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Mixing at the Batchelor Scale for White-In-Time Flows
We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from…
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A Dynamical Systems Perspective on the Analysis of Neural Networks
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements.…
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Characterizing Dynamical Stability of Stochastic Gradient Descent in Overparameterized Learning
For overparameterized optimization tasks, such as those found in modern machine learning, global minima are generally not unique. In order to understand generalization in these settings, it is vital to study to which minimum an optimization algorithm converges. The possibility of having…
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Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
Published · Probability theory and related fields, 192 (2025) 3/4, pp. 1447-1515. The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a…