Harnack Inequality for Functional SDEs with Bounded Memory

Abdelhadi Es-Sarhir (TU (Berlin))
Max-K. von Renesse (TU (Berlin))
Michael Scheutzow (TU (Berlin))

Abstract


We use a coupling method for functional stochastic differential equations with bounded memory to establish an analogue of Wang's dimension-free Harnack inequality [13]. The strong Feller property for the corresponding segment process is also obtained.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 560-565

Publication Date: December 13, 2009

DOI: 10.1214/ECP.v14-1513

References

  1. Arnaudon, Marc; Thalmaier, Anton; Wang, Feng-Yu. Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130 (2006), no. 3, 223--233. MR2215664 (2007i:58032)
  2. Da Prato, Giuseppe; Röckner, Michael; Wang, Feng-Yu. Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257 (2009), no. 4, 992--1017. MR2535460
  3. Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems.London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7 MR1417491 (97k:60165)
  4. A. Es--Sarhir, O. van Gaans and M.Scheutzow, Invariant measures for stochastic delay equations with superlinear drift term, Diff. Int. Eqs 23 (1-2) (2010), 189--200.
  5. M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Prob. Theory Rel. Fields, DOI 10.1007/s00440-009-0250-6.
  6. Kassmann, Moritz. Harnack inequalities: an introduction. Bound. Value Probl. 2007, Art. ID 81415, 21 pp. MR2291922 (2007j:35001)
  7. Krylov, N. V. A simple proof of the existence of a solution to the Itô equation with monotone coefficients.(Russian) Teor. Veroyatnost. i Primenen. 35 (1990), no. 3, 576--580; translation in Theory Probab. Appl. 35 (1990), no. 3, 583--587 (1991) MR1091217 (92m:60047)
  8. Lifshits, M. A. Gaussian random functions.Mathematics and its Applications, 322. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp. ISBN: 0-7923-3385-3 MR1472736 (98k:60059)
  9. W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ. 9 (2009), 747–-770.
  10. M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ, to appear.
  11. Scheutzow, Michael. Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5 (2005), no. 2, 163--174. MR2147280 (2006b:60130)
  12. Scheutzow, M. Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12 (1984), no. 1, 41--80. MR0738934 (85h:60087)
  13. Wang, Feng-Yu. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 (1997), no. 3, 417--424. MR1481127 (98i:58253)
  14. Wang, Feng-Yu. Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35 (2007), no. 4, 1333--1350. MR2330974 (2008e:60192)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.