Published · Real Analysis Exchange 48(1), 101–106 (2023). In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb{R}^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb{R}^n$ is never…
Published · Journal of Differential Equations 392, 74–127 (2024). We provide generalizations of a class of stochastic Gronwall inequalities that has been studied by von Renesse and Scheutzow (2010), Scheutzow (2013), Xie and Zhang (2020) and Mehri and Scheutzow (2021). This…
Published · Journal of Differential Equations 389, 114–149 (2024). We consider in a smooth and bounded two dimensional domain the convergence in the $L^2$ norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding…
Published · Advances in Continuous and Discrete Models (2022). We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover non-linear variational problems such as the $p$-Laplace equation or the Modica-Mortola energy with essential or natural…
Published · Stochastic Processes and their Applications 157, 376–412 (2023). We construct and analyze the Jacobi process – in mathematical biology referred to as Wright-Fisher diffusion – using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space…
Published · International Conference on Learning Representations (ICLR 2022). We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted…
Published · The annals of probability, 50 (2022) 6, pp. 2288-2343. We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in…
Published · Mathematics in Engineering 4(4), 1–21 (2022).
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…
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…
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…
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…
