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Harnack inequality and applications for stochastic evolution equations with monotone drifts. (English) Zbl 1239.60058
Summary: As a Generalization to [F.-Y. Wang, Ann. Probab. 35, No. 4, 1333–1350 (2007; Zbl 1129.60060)] where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic $$p$$-Laplace equation in Hilbert space.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J35 Transition functions, generators and resolvents 47D07 Markov semigroups and applications to diffusion processes 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35R50 PDEs of infinite order 60F10 Large deviations 60H40 White noise theory
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