Benjamin Gess

Random attractors for a class of stochastic partial differential equations driven by general additive noise

Abstract The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction–diffusion equations, the stochastic p -Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time fractional Brownian motion and space-time Levy noise as admissible random perturbations. Moreover, cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained.

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L 2-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space-time rough signals is proven for initial data in L 1 (O) on bounded domains O. The generation of a continuous, order-preserving random dynamical system (RDS) on L 1 (O) and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in L ∞ (O) norm. Uniform L ∞ bounds and uniform space-time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated. A pathwise Wong-Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space-time rough signals existence of solutions is proven for initial data in L m+1 (O).

Synchronization by noise

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 with additive noise, these conditions are also essentially necessary. In addition, we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point. As a result, synchronization by noise is proven for a large class of SDE with additive noise. In particular, we prove that the random attractor for an SDE with drift given by a (multidimensional) double-well potential and additive noise consists of a single random point. All examples treated in Tearne (Probab Theory Relat Fields 141(1–2):1–18, 2008) are also included.

RANDOM ATTRACTORS FOR DEGENERATE STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

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 the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate

Multi-valued, singular stochastic evolution inclusions

p

Stability of solutions to stochastic partial differential equations

-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate