Stochastic dynamics asks how random systems evolve over long times and how individual sample paths organize into coherent behaviour.
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…
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…
Published · Annals of Applied Probability 36(2), 1690–1736 (2026). We derive weak and strong well-posedness for one-dimensional stochastic differential equations with measure-valued drift driven by a symmetric stable process…
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…
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…
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…
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…
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…
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…
The dynamics of the solutions to a class of conservative SPDEs are analysed from two perspectives: Firstly, a probabilistic construction of a corresponding random dynamical system is given for the first time. Secondly, the existence and uniqueness of invariant measures, as well…
