Journal of Evolution Equations

, 9:747

Harnack inequality and applications for stochastic evolution equations with monotone drifts

Article

DOI: 10.1007/s00028-009-0032-8

Cite this article as:
Liu, W. J. Evol. Equ. (2009) 9: 747. doi:10.1007/s00028-009-0032-8

Abstract

As a Generalization to Wang (Ann Probab 35:1333–1350, 2007) 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.

Mathematics Subject Classification (2000)

60H15 60J35 47D07 

Keywords

Stochastic evolution equation Harnack inequality Strong Feller property Ergodicity Spectral gap p-Laplace equation Porous media equation 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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