Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

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 · 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 · 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 · 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 · Journal de mathématiques pures et appliquées, 101 (2014) 6, pp. 789-827. We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular…