Benjamin Gess

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).

Published · Probability theory and related fields, 176 (2020) 1/2, pp. 421-448. We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range…

Published · Journal of differential equations, 266 (2019) 6, pp. 3732-3763. We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well – posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows…

Published · Annales de l’Institut Henri Poincaré / B : Probabilites et statistiques, 55 (2019) 3, pp. 1765-1796. We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $arepsilon$ and thus imply analogous estimates for…

Published · Archive for rational mechanics and analysis, 233 (2019) 1, pp. 249-322. We prove the pathwise well-posedness of stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. As a consequence, the generation of a random dynamical system is obtained. This extends…

Published · Journal of theoretical probability, 33 (2020) 2, pp. 611-648. We consider finite dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution…