- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in Mathematical Physics
Volume 2014 (2014), Article ID 479873, 10 pages
Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise
1Department of Mathematics and Physics, Bengbu College, 1866 Caoshan Road, Bengbu, Anhui 233030, China
2School of Mathematics and Statistics, Nanjing Audit University, 86 West Yu Shan Road, Pukou, Nanjing 211815, China
Received 21 September 2013; Revised 27 December 2013; Accepted 27 December 2013; Published 12 February 2014
Academic Editor: Ming Li
Copyright © 2014 Xichao Sun and Junfeng Liu. 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.
- J. Droniou and C. Imbert, “Fractal first order partial differential equations,” Archive for Rational Mechanics and Analysis, vol. 182, pp. 229–261, 2004.
- V. V. Uchaikin and V. M. Zolotarev, Chance and Stability, Stable Distributions and Their Applications, Modern Probability and Statistics, 1999.
- L. Chen, Y. Chai, R. Wu, J. Sun, and T. Ma, “Cluster synchronization in fractional-order complex dynamical networks,” Physics Letters A, vol. 376, no. 35, pp. 2381–2388, 2010.
- M. Li, W. Zhao, and C. Cattani, “Delay bound: fractal traffic passes through network servers,” Mathematical Problems in Engineering, vol. 2013, Article ID 157636, 15 pages, 2013.
- M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.
- S. Cohen and A. Lindner, “A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process,” Journal of Statistical Planning and Inference, vol. 143, no. 8, pp. 1295–1306, 2013.
- K. Jaczak-Borkowska, “Generalized BSDEs driven by fractional Brownian motion,” Statistics and Probability Letters, vol. 83, no. 3, pp. 805–811, 2013.
- M. Besal and C. Rovira, “Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter ,” Stochastics and Dynamics, vol. 12, no. 4, pp. 1–24, 2012.
- D. T. Nguyen, “Mackey-Glass equation driven by fractional Brownian motion,” Physica A, vol. 391, no. 22, pp. 5465–5472, 2012.
- Y. S. Mishura and S. V. Posashkova, “Stochastic differential equations driven by a Wiener process and fractional Brownian motion: convergence in Besov space with respect to a parameter,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1166–1180, 2011.
- M. Li and W. Zhao, “Quantitatively investigating locally weak stationarity of modified multifractional Gaussian noise,” Physica A, vol. 391, no. 24, pp. 6268–6278, 2012.
- M. Li, Y. Q. Chen, J. Y. Li, and W. Zhao, “Hölder scales of sea level,” Mathematical Problems in Engineering, vol. 2012, Article ID 863707, 22 pages, 2012.
- C. Mueller, “The heat equation with Lévy noise 1,” Stochastic Processes and Their Applications, vol. 74, no. 1, pp. 67–82, 1998.
- J. Wu, “Fractal Burgers equation with stable Lévy noise,” in Proceedings of the 7th International Conference SPDE and Applications, January 2004.
- T. Chang and K. Lee, “On a stochastic partial differential equation with a fractional Laplacian operator,” Stochastic Processes and Their Applications, vol. 122, no. 9, pp. 3288–3311, 2012.
- A. Truman and J. Wu, “Fractal Burgers equation driven by Lévy noises,” in Stochastic Partial Differential Equations and Applications, G. da Prato and L. Tubaro, Eds., vol. 245 of Lecture Notes in Pure and Applied Mathematics, pp. 295–310, Chapman Hall/CRC Taylor, 2006.
- J. Liu, L. Yan, and Y. Cang, “On a jump-type stochastic fractional partial differential equation with fractional noises,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 16, pp. 6060–6070, 2012.
- D. Wu, “On the solution process for a stochastic fractional partial differential equation driven by space-time white noise,” Statistics and Probability Letters, vol. 81, no. 8, pp. 1161–1172, 2011.
- M. S. Taqqu, “Weak convergence to fractional brownian motion and to the rosenblatt process,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 31, no. 4, pp. 287–302, 1975.
- R. Delgado and M. Jolis, “Weak approximation for a class of Gaussian processes,” Journal of Applied Probability, vol. 37, no. 2, pp. 400–407, 2000.
- X. Bardina, M. Jolis, and L. Quer-Sardanyons, “Weak convergence for the stochastic heat equation driven by Gaussian white noise,” Electronic Journal of Probability, vol. 15, pp. 1267–1295, 2010.
- B. Boufoussi and S. Hajji, “An approximation result for a quasi-linear stochastic heat equation,” Statistics and Probability Letters, vol. 80, no. 17-18, pp. 1369–1377, 2010.
- T. E. Mellall and Y. Ouknine, “Weak convergence for qualinear stochastic heat equation driven by a fractional noise with Hurst parameter ,” Stochastics and Dynamics, vol. 13, no. 3, pp. 1–13, 2013.
- L. Debbi, “On some properties of a high order. Fractional differential operator which is not in general selfadjoint,” Applied Mathematical Sciences, vol. 1, pp. 1325–1339, 2007.
- L. Debbi and M. Dozzi, “On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension,” Stochastic Processes and Their Applications, vol. 115, no. 11, pp. 1764–1781, 2005.
- J. B. Walsh, “An introduction to stochastic partial differential equations,” in École d'Été de Probabilités de Saint Flour XIV—1984, vol. 1180 of Lecture Notes in Mathematics, pp. 266–439, Springer, Berlin, Germany, 1986.
- D. Nualart and Y. Ouknine, “Regularization of quasilinear heat equation by a fractional noise,” Stochastics and Dynamics, vol. 4, no. 2, pp. 201–221, 2004.
- T. Komatsu, “On the martingale problem for generators of stable processes with perturbations,” Osaka Journal of Mathematics, vol. 21, pp. 113–132, 1984.