Published · Stochastic processes and their applications, 127 (2017) 9, pp. 2961-3004. We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of…
Published · Probability theory and related fields, 173 (2019) 3/4, pp. 1063-1098. We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed…
Published · SIAM journal on numerical analysis, 54 (2016) 4, pp. 2187-2209. We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier’s transport-collapse algorithm and uses characteristics defined in the setting…
Published · Discrete and Continuous Dynamical Systems B 21(9), 3003–3014 (2016). We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of…
Published · SIAM journal on mathematical analysis, 48 (2016) 6, pp. 4094-4125. Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, Tölle; J. Math. Pures Appl.,…
Published · Journal of differential equations, 260 (2016) 6, pp. 4973-5025. We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift…
Published · The annals of probability, 45 (2017) 2, pp. 1325-1350. We provide sufficient conditions for weak synchronization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting…
Published · SIAM journal on mathematical analysis, 47 (2015) 5, pp. 4058-4090. We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the Bak-Tang-Wiesenfeld model for self-organized criticality. A…
Published · Communications on pure and applied mathematics, 70 (2017) 8, pp. 1562-1597. We study the long-time behavior and the regularity of pathwise entropy solutions to stochastic scalar conservation laws with random in time spatially homogeneous fluxes and periodic initial data. We prove that the…
Published · Probability theory and related fields, 168 (2017) 3/4, pp. 511-556. We provide sufficient conditions for synchronization by noise, i.e. under these conditions we prove that weak random attractors for random dynamical systems consist of single random points. In the case of SDE…
Published · Communications in mathematical sciences, 13 (2015) 6, pp. 1569-1597. We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent…
Published · Transactions of the American Mathematical Society, 369 (2017) 5, pp. 3017-3045. The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous…
