Heat diffusion in a channel under white noise modeling of turbulence
Published · Mathematics in Engineering 4(4), 1–21 (2022).
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Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise
Published · Mathematics of computation, 92 (2023), pp. 1931-1976. In micro-fluidics not only does capillarity dominate but also thermal fluctuations become important. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height…
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Well-posedness of the Dean–Kawasaki and the nonlinear Dawson–Watanabe equation with correlated noise
Published · Archive for rational mechanics and analysis, 248 (2024) 2, p. 20. In this paper we prove the well-posedness of the generalized Dean–Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only…
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Martingale solutions to the stochastic thin-film equation in two dimensions
Published · Annales de l’Institut Henri Poincaré B 60(1), 373–412 (2024). We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension $d=2$ and allow in particular for solutions with non-full support. The construction…
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Notes on Exact Boundary Values in Residual Minimisation
Published · Mathematical and Scientific Machine Learning (MSML 2022). We analyse the difference in convergence mode using exact versus penalised boundary values for the residual minimisation of PDEs with neural network type ansatz functions, as is commonly done in the context…
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Stochastic partial differential equations arising in self-organized criticality
Published · The annals of applied probability, 35 (2025) 1, pp. 481-522. We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation…
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Stabilization by transport noise and enhanced dissipation in the Kraichnan model
Published · Journal of evolution equations, 25 (2025) 2, p. 42. Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown…
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Deep Learning Schemes For Parabolic Nonlocal Integro-Differential Equations
Published · Partial Differential Equations and Applications 3 (2022). In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been…
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Error Estimates for the Deep Ritz Method with Boundary Penalty
Published · Mathematical and Scientific Machine Learning (MSML 2022). We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method…
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Sharpness of Lenglart’s domination inequality and a sharp monotone version
Published · Electronic Communications in Probability 26, 1–8 (2021). We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\, p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and…
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Numerical approximation of singular-degenerate parabolic stochastic PDEs
Published · IMA journal of numerical analysis, 44 (2024) 4, pp. 2090-2137. We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a…
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An Example of Intrinsic Randomness in Deterministic PDEs
Published · Stochastics and dynamics, 22 (2022) 7, p. 2240023. A new mechanism leading to a random version of Burgers’ equation is introduced: it is shown that the Totally Asymmetric Exclusion Process in discrete time (TASEP) can be understood as an intrinsically…