Feine Eigenschaften von stochastischen Evolutionsgleichungen und deren Anwendungen

Liu W. Fine properties of stochastic evolution equations and their applications. Bielefeld (Germany): Bielefeld University; 2009.In this work, we aim to study some fine properties for a class of nonlinear SPDE within the variational framework. The results consist of three main parts. In the first part, we study the asymptotic behavior of nonlinear SPDE with small multiplicative noise. A Freidlin-Wentzell large deviation principle is established for the distributions of solutions to a large class of SPDE, which include all stochastic evolution equations with monotone coefficients. In the second part, some properties of invariant measures and transition semigroups are investigated for SPDE with additive noise. The main tool is the dimension-free Harnack inequality, which is established by using a coupling method and Girsanov transformation techniques. Subsequently, the Harnack inequality is used to derive the ergodicity, compactness and contractivity (e.g. hyperboundedness or ultraboundedness) for the associated transition semigroups. Moreover, the uniformly exponential convergence of the transition semigroup to the invariant measure and the existence of a spectral gap are also obtained. These results are first established for general stochastic evolution equations with strongly dissipative drift, e.g. stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation (p >= 2) in Hilbert space. Stochastic fast diffusion equations and the singular stochastic p-Laplace equation (1 < p < 2) are investigated separately by using more delicate arguments due to the weak dissipativity of the drifts. In the last part, the invariance of subspaces under the solution flow of SPDE is investigated. We prove that the solution of an SPDE takes values in some suitable subspace of the state space if the initial state does so. This gives the stronger regularity estimates for the solution of an SPDE, which can be used for further study of the corresponding random dynamical system. As examples, the main results are applied to many concrete SPDEs in Hilbert space