Published · Workshop on Integration of Deep Neural Models and Differential Equations at ICLR 2020. Residual networks (ResNets) are a deep learning architecture that substantially improved the state of the art performance in certain supervised learning tasks. Since then, they have received continuously growing attention. ResNets have…
Published · Stochastics and Dynamics 20(6), 2040006 (2020). We prove uniform synchronisation by noise with rates for the stochastic quantisation equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the…
Published · Journal of differential equations, 269 (2020) 4, pp. 3414-3455. We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable…
Published · SIAM journal on mathematical analysis, 52 (2020) 5, pp. 4524-4564. The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates the existence and uniqueness of invariant measures…
Published · Stochastic processes and their applications, 130 (2020) 12, pp. 7260-7302. We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition…
Published · Nonlinear analysis / A, 187 (2019), pp. 259-278. The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting…
Published · Journal of machine learning research, 21 (2020) 136, pp. 1-48. We prove the local convergence to minima and estimates on the rate of convergence for the stochastic gradient descent method in the case of not necessarily globally convex nor contracting objective functions…
Published · Analysis and PDE, 13 (2020) 8, pp. 2441-2480. Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is…
Published · Electronic journal of probability, 25 (2020), p. 35. We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the…
Published · BIT : numerical mathematics, 60 (2020) 4, pp. 1057-1073. Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence…
Published · Journal de mathématiques pures et appliquées, 148 (2021), pp. 221-266. We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves…
Published · Stochastic Partial Differential Equations and Related Fields, Springer Proceedings in Mathematics & Statistics 229, pp. 43–67 (2018).
