Random attractors for singular stochastic evolution equations

The existence of random attractors for singular stochastic evolution equations (SEE) perturbed by general additive noise is proven. The drift is only assumed to satisfy the standard assumptions of the variational approach to SEE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamical systems it is proven that the random attractor consists of a single random point. In case of real, linear, multiplicative noise finite time extinction is obtained. Applications include stochastic generalized fast diffusion equations and stochastic generalized singular p-Laplace equations perturbed by Lévy noise with jump measure having finite first and second moments.