Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 601327, 7 pages
http://dx.doi.org/10.1155/2014/601327
Research Article

Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

1School of Sciences, South China University of Technology, Guangzhou 510640, China
2School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China
3Department of Mathematics, Guangdong University of Education, Guangzhou 510310, China

Received 22 August 2013; Accepted 27 October 2013; Published 15 January 2014

Academic Editors: T. Prieto-Rumeau and K. Saito

Copyright © 2014 Caibin Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. C. G. Rogers, “Arbitrage with fractional brownian motion,” Mathematical Finance, vol. 7, no. 1, pp. 95–105, 1997. View at Google Scholar · View at Scopus
  2. L. Decreusefond and A. S. Üstünel, “Stochastic analysis of the fractional Brownian motion,” Potential Analysis, vol. 10, no. 2, pp. 177–214, 1999. View at Google Scholar · View at Scopus
  3. S. J. Lin, “Stochastic analysis of fractional Brownian motions,” Stochastics and Stochastic Reports, vol. 55, no. 1-2, pp. 121–140, 1995. View at Google Scholar
  4. W. Dai and C. C. Heyde, “Itô's formula with respect to fractional Brownian motion and its application,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 4, pp. 439–448, 1996. View at Google Scholar · View at Scopus
  5. T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000. View at Google Scholar · View at Scopus
  6. E. Alòs, O. Mazet, and D. Nualart, “Stochastic calculus with respect to Gaussian processes,” Annals of Probability, vol. 29, no. 2, pp. 766–801, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. P. Carmona, L. Coutin, and G. Montseny, “Stochastic integration with respect to fractional Brownian motion,” Annales de l'institut Henri Poincare B, vol. 39, no. 1, pp. 27–68, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. R. J. Elliott and J. Van Der Hoek, “A general fractional white noise theory and applications to finance,” Mathematical Finance, vol. 13, no. 2, pp. 301–330, 2003. View at Publisher · View at Google Scholar · View at Scopus
  9. F. Biagini, B. Øksendal, A. Sulem, and N. Wallner, “An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion,” Proceedings of the Royal Society A, vol. 460, no. 2041, pp. 347–372, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. D. Nualart and A. Răşcanu, “Differential equations driven by fractional Brownian motion,” Collectanea Mathematica, vol. 53, no. 1, pp. 55–81, 2002. View at Google Scholar
  11. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer, Berlin, Germany, 1998.
  12. B. Maslowski and B. Schmalfuss, “Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion,” Stochastic Analysis and Applications, vol. 22, no. 6, pp. 1577–1607, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. M. J. Garrido-Atienza, K. Lu, and B. Schmalfuss, “Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion,” Discrete and Continuous Dynamical Systems B, vol. 14, no. 2, pp. 473–493, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. M. J. Garrido-Atienza, B. Maslowski, and B. Schmalfuß, “Random attractors for stochastic equations driven by a fractional brownian motion,” International Journal of Bifurcation and Chaos, vol. 20, no. 9, pp. 2761–2782, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. M. J. Garrido-Atienza, K. Lu, and B. Schmalfuß, “Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion,” Journal of Differential Equations, vol. 248, no. 7, pp. 1637–1667, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. T. Caraballo, M. J. Garrido-Atienza, and T. Taniguchi, “The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 11, pp. 3671–3684, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. N. V. Krylov and B. L. Rozovskiĭ, “Stochastic evolution equations,” Seriya Sovremennye Problemy Matematiki, vol. 14, pp. 71–146, 1979, Translated from Itogi Nauki i Tekhniki. View at Google Scholar
  18. C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, vol. 1905 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2007.
  19. B. Gess, W. Liu, and M. Röckner, “Random attractors for a class of stochastic partial differential equations driven by general additive noise,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1225–1253, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Kunita, Stochastic Flows and Stochastic Differential Equations, vol. 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
  21. T. E. Duncan, B. Pasik-Duncan, and B. Maslowski, “Fractioanl Brownian motion and stochastic equations in Hilbert spaces,” Stochastics and Dynamics, vol. 2, no. 2, pp. 225–250, 2002. View at Google Scholar
  22. Y. Wang, Variational solutions to SPDE perturbed by a general Gaussian noise [Ph.D. thesis], Purdue University, 2009.
  23. W. Liu and M. Röckner, “SPDE in Hilbert space with locally monotone coefficients,” Journal of Functional Analysis, vol. 259, no. 11, pp. 2902–2922, 2010. View at Publisher · View at Google Scholar · View at Scopus