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…
Published · Stochastics and partial differential equations : analysis and computations, 11 (2023) 1, pp. 211-268. The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong…
Published · Journal de mathématiques pures et appliquées, 128 (2019), pp. 225-263. The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the…
Published · Journal of the European Mathematical Society, 23 (2021) 2, pp. 425-465. We prove optimal regularity for solutions to porous media equations in Sobolev spaces, based on velocity averaging techniques. In particular, the obtained regularity is consistent with the optimal regularity in the linear…
Published · Journal of functional analysis, 277 (2019) 5, pp. 1469-1498. We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results…
Published · Annales de l’Institut Henri Poincaré / C, 36 (2019) 2, pp. 505-521. We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced,…
Published · Communications in partial differential equations, 43 (2018) 12, pp. 1702-1736. We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are…
Published · The annals of probability, 46 (2018) 5, pp. 2495-2544. We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging…
