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…
Published · Annales de l’Institut Henri Poincaré / C, 33 (2016) 4, pp. 933-963. We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space-time transport noise and affine-linear noise driven by a geometric p-rough path. In particular,…
Published · Communications in mathematical physics, 335 (2015) 1, pp. 309-344. We prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem posed and left…
Published · The annals of probability, 44 (2016) 1, pp. 684-738. We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and…
It is a well-known fact that finite rho-variation of the covariance (in 2D sense) of a general Gaussian process implies finite rho-variation of Cameron-Martin paths. In the special case of fractional Brownian motion (think: 2H=1/rho), in the rougher than Brownian…
We present sufficient conditions for finite controlled rho-variation of the covariance of Gaussian processes with stationary increments, based on concavity or convexity of their variance function. The motivation for this type of conditions comes from recent work of Hairer [CPAM,2011].…
Published · SIAM journal on mathematical analysis, 45 (2013) 5, pp. 2734-2766. We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies…
Published · Journal of dynamics and differential equations, 25 (2013) 1, pp. 121-157. We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only…
