Stochastic Functional (Partial) Differential Equations

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


In this chapter we investigate Harnack/shift Harnack inequalities and derivative formulas for stochastic functional differential equations. In this case, the strong or mild solution is no longer Markovian. These inequalities and formulas are therefore established for the semigroup associated with the functional (or segment) solutions. To this end, a time larger than the length of delay is necessary in order to construct a successful coupling by change of measure. Based on Bao et al. (Derivative formula and Harnack inequality for degenerate functional SDEs, to appear in Stochast. Dynam.; Bismut Formulae and Applications for Functional SPDEs, to appear in Bull. Math. Sci.), Shao et al. (Electron. J. Probab. 17:1–18, 2012), Wang and Yuan (Stoch. Process. Their Appl. 121:2692–2710, 2011), several specific models of elliptic SDDEs, semilinear SDPDEs, and degenerate SDDEs are considered.


Brownian Motion Mild Solution Harnack Inequality Coupling Time Malliavin Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 4.
    M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient estimates and Harnack inequalities on noncompact Riemannian manifolds, Stochastic Process. Appl. 119(2009), 3653–3670.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 7.
    J. Bao, F.-Y. Wang, C. Yuan, Derivative formula and Harnack inequality for degenerate functional SDEs, to appear in Stochastics and Dynamics. Stoch. Dyn. 13(2013), 1250013, 22 pages.Google Scholar
  3. 8.
    J. Bao, F.-Y. Wang, C. Yuan, Bismut Formulae and Applications for Functional SPDEs, to appear in Bull. Math. Sci. Bull. Sci. Math. 137(2013), 509–522.MathSciNetCrossRefGoogle Scholar
  4. 15.
    A. Es-Sarhir, M.-K. v. Renesse, M. Scheutzow, Harnack inequality for functional SDEs with bounded memory, Electron. Commun. Probab. 14(2009), 560–565.Google Scholar
  5. 18.
    S. Fang, T. Zhang, A study of a class of stochastic differential equations with nonLipschitzian coefficients, Probab. Theory Related Fields, 132(2005), 356–390.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 19.
    A. Guillin, F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differential Equations 253(2012), 20–40.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 22.
    N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (Second Edition), North-Holland, Amsterdam, 1989.Google Scholar
  8. 25.
    G. Q. Lan, Pathwise uniqueness and nonexplosion of SDEs with nonLipschitzian coefficients, Acta Math. Sinica (Chin. Ser.) 52(2009), 731–736.MathSciNetzbMATHGoogle Scholar
  9. 33.
    X. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, 1997.zbMATHGoogle Scholar
  10. 44.
    M.-K. von Renesse, M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ. 18(2010), 267–284.MathSciNetzbMATHGoogle Scholar
  11. 45.
    T. Seidman,How violent are fast controls? Math. of Control Signals Systems, 1(1988), 89–95.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 47.
    J. Shao, F.-Y. Wang. C. Yuan, Harnack inequalities for stochastic (functional) differential equations with nonLipschitzian coefficients, Elec. J. Probab. 17(2012), 1–18.MathSciNetGoogle Scholar
  13. 48.
    T. Taniguchi, The existence and asymptotic behaviour of solutions to nonLipschitz stochastic functional evolution equations driven by Poisson jumps, Stochastics 82(2010), 339–363.MathSciNetzbMATHGoogle Scholar
  14. 70.
    F.-Y. Wang, X. Zhang, Derivative formula and applications for degenerate diffusion semigroups, to appear in J. Math. Pures Appl. 99(2013), 726–740.MathSciNetGoogle Scholar
  15. 71.
    T. Yamada, S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11(1971), 155–167.MathSciNetzbMATHGoogle Scholar
  16. 75.
    X.-C. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl. 120(2010), 1929–1949.MathSciNetzbMATHCrossRefGoogle 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

Personalised recommendations