Stochastic Dynamics
Mathematical challenges
Stochastic dynamics asks how random systems evolve over long times and how individual sample paths organize into coherent behaviour. We study random dynamical systems generated by stochastic differential equations and SPDEs.
Our questions include ergodicity, mixing, stability and the formation of random attractors. Particular attention is paid to synchronization by noise, where common fluctuations can bring initially different states together, and to regularization mechanisms that change the qualitative dynamics. We combine probabilistic, analytical and geometric techniques to identify Lyapunov behaviour, quantify convergence and understand how microscopic stochasticity affects macroscopic stability. These tools also inform the study of learning algorithms and large interacting systems.
Current directions
- Random dynamical systems generated by SPDEs
- Synchronization by noise
- Mixing and ergodicity
- Random attractors and stability
- Long-time qualitative behaviour
Related people
Related projects
Selected publications
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Non-Asymptotic Analysis of Projected Gradient Descent for Physics-Informed Neural Networks
Published · Scientific Machine Learning: Emerging Topics, SEMA SIMAI Springer Series (2026). In this work, we provide a non-asymptotic convergence analysis of projected gradient descent for physics-informed neural networks for the Poisson equation. Under suitable assumptions, we show that the optimization error can be…
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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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Low temperature expansion for the Euclidean $Φ^4_2$-measure
Published · Transactions of the American Mathematical Society, (2026). We study asymptotic expansions of the Euclidean $Φ^4_2$-measure in the low-temperature regime. In particular, this extends the asymptotic expansions of Gaussian function space integrals developed in Schilder (1966) and Ellis and Rosen…
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Stochastic Modified Flows for Riemannian Stochastic Gradient Descent
Published · SIAM journal on control and optimization, 62 (2024) 6, pp. 3288-3314. We give quantitative estimates for the rate of convergence of Riemannian stochastic gradient descent (RSGD) to Riemannian gradient flow and to a diffusion process, the so-called Riemannian stochastic modified flow (RSMF). Using…
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Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent
Published · Journal of machine learning research, 25 (2024) 30, pp. 1-27. We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called…
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Exponential convergence rates for momentum stochastic gradient descent in the overparametrized setting
Published · Mathematical programming, (2026). We prove explicit bounds on the exponential rate of convergence for the momentum stochastic gradient descent scheme (MSGD) for arbitrary, fixed hyperparameters (learning rate, friction parameter) and its continuous-in-time counterpart in the…
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Long-time behaviour of stochastic Hamilton-Jacobi equations
Published · Journal of functional analysis, 286 (2024) 4, p. 110269. The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them…
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Lyapunov exponents and synchronisation by noise for systems of SPDEs
Published · The annals of probability, 52 (2024) 5, pp. 1903-1953. Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the…