Semilinear Stochastic Partial Differential Equations

  • Feng-Yu Wang
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we establish Harnack/shift Harnack inequalities and derivative formulas for the semigroup associated with mild solutions of semilinear stochastic differential equations on Hilbert spaces. For simplicity, we consider only single-valued equations with a time-homogeneous linear operator; see Da Prato et al. (J. Funct. Anal. 257:992–1017, 2009), Ouyang (Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14:261–278, 2011), and Zhang (Probab. Lett. 83:1184–1192, 2013; Shift Harnack inequality and integration by part formula for semilinear SPDE, arXiv:1208.2425) for a study of equations possibly with a time-dependent linear operator and a multivalued nonlinear term. In the first section we introduce finite-dimensional approximations of mild solutions, which will then be used in the other sections to derive results from existing ones in finite dimensions. Materials in this chapter are modified from Wang (Ann. Probab. 39:1449–1467, 2011; Integration by parts formula and shift Harnack inequality for stochastic equations, arXiv:1203.4023), Wang and Zhang (Log-Harnack inequality for mild solutions of SPDEs with strongly multiplicative noise, arXiv:1210.6416).


  1. 10.
    G. Da Prato, M. Röckner, F.-Y. Wang, Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. Funct. Anal. 257(2009), 992–1017.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 11.
    G. Da Prato, J. Zabczyk, Stochastic Equations In Infinite Dimensions, Cambridge University Press, Cambridge, 1992.zbMATHCrossRefGoogle Scholar
  3. 12.
    G. Da Prato, J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, Cambridge University Press, Cambridge, 1996.zbMATHCrossRefGoogle Scholar
  4. 36.
    S.-X. Ouyang, Harnack inequalities and applications for multivalued stochastic evolution equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(2011), 261–278.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 72.
    S.-Q. Zhang, Harnack inequality for semilinear SPDEs with multiplicative noise, Statist. Probab. Lett. 83(2013), 1184–1192.zbMATHCrossRefGoogle Scholar
  6. 73.
    S.-Q. Zhang, Shift Harnack inequality and integration by part formula for semilinear SPDE, arXiv:1208.2425.Google Scholar
  7. 74.
    T. Zhang, White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities, Potential Anal. 33(2010), 137–151.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Feng-Yu Wang
    • 1
    • 2
  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina, People’s Republic
  2. 2.Department of MathematicsSwansea UniversitySwanseaUK

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