Publications
The catalogue keeps one record per scientific work. A preprint record is updated with journal information rather than duplicated when a journal version appears.
Display scientific works by current members that first became publicly available while the author was a member of SAiS.
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- Benjamin Gess
- Max Sauerbrey
- Adrian Martini
- Paul Nikolaev
- Johannes Müller
- Sarah Geiss
- Dennis Chemnitz
- Javier Castro
- Shanshan Hu
- Thomas Müller
- Tom Hapke
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Mathematical challenges
Publication catalogue
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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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SVI solutions to stochastic nonlinear diffusion equations on general measure spaces
Published · Journal of evolution equations, 24 (2024) 4, p. 94. We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a…
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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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Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations
A broad class of possibly non-unique generalized kinetic solutions to hyperbolic-parabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev…
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Landau-Lifshitz-Navier-Stokes Equations: Large Deviations and Relationship to The Energy Equality
Accepted / forthcoming · Annals of Applied Probability. The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the consistency of the large…
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Solutions to the stochastic thin-film equation for initial values with non-full support
Published · Transactions of the American Mathematical Society, 379 (2026) 4, pp. 2343-2383. The stochastic thin-film equation with mobility exponent $n\in [\frac{8}{3},3)$ on the one-dimensional torus with multiplicative Stratonovich noise is considered. We show that martingale solutions exist for non-negative initial values. This advances on…
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The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion
Published · Communications on pure and applied mathematics, 78 (2025) 9, pp. 1609-1655. The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied…
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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…
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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…