On the stochastic two-component b-family system

References

• Guha, P. 2007. Euler-Poincaré formalism of (two component) Degasperis-Procesi and Holm-Staley type systems. J. Nonlinear Math. Phys. 14:398–429.
   Web of Science ®Google Scholar
• Constantin, A., and Ivanov, R. 2008. On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A 372:7129–7132.
   Web of Science ®Google Scholar
• Dullin, H. R., Gottwald, G. A., and Holm, D. D. 2003. Camassa-Holm, Korteweg-de Veris-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dynam. Res. 33:73–79.
   Web of Science ®Google Scholar
• Dullin, H. R., Gottwald, G. A., and Holm, D. D. 2004. On asymptotically equivalent shallow water wave equations. Physica D 190:1–14.
   Web of Science ®Google Scholar
• Camassa, R., and Holm, D. 1993. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1661–1664.
   PubMed Web of Science ®Google Scholar
• Fuchssteiner, B., and Fokas, A. 1981. Symplectic structures, their Bäklund transformations and hereditary symmetries. Phys. D 4:47–66.
   Web of Science ®Google Scholar
• Degasperis, A., and Procesi, M. 1999. Asymptotic integrability. In: Symmetry and Perturbation Theory, Rome, 1998. River Edge, NJ: World Science, 23–37.
   Google Scholar
• Chen, Y., Gao, H. J. and Liu, Y. 2013. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete Contin. Dyn. Syst. 33:3407–3441.
   Web of Science ®Google Scholar
• de Bouard, A., Debussche, A., and Tsutsumi, Y. 1999. White noise driven Korteweg-de Vries equation. J Funct Anal 169:532–558.
   Web of Science ®Google Scholar
• Chen, Y., Gao, H. J., and Guo, B. L. 2012. Well posedness for the stochastic Camassa-Holm equation. J. Differential Equations 253:2353–2379.
   Web of Science ®Google Scholar
• Varadhan, S. R. S. 1984. Large Deviations and Applications. Philadelphia: SIAM.
   Google Scholar
• Azencott, R. 1980. Grandes déviations et applications. École dété Probabilités de Saint-Flour VIII-1978. Lecture Notes in Mathematics, 779. Berlin: Springer, 1–176.
   Google Scholar
• Freidlin, M. I., and Wentzell, A. D. 1984. Random Perturbations of Dynamical Systems. New York: Springer.
   Google Scholar
• Chenal, F. and Millet, A. 1997. Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72:161–186.
   Web of Science ®Google Scholar
• Chow, P. L. 1992. Large deviation problem for some parabolic Itô equations. Comm. Pure Appl. Math. 45:97–120.
   Web of Science ®Google Scholar
• Sritharan, S. S., and Sundar, P. 2006. Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process. Appl. 116:1636–1659.
   Web of Science ®Google Scholar
• Dupuis, P., and Ellis, R. S. 1997. A Weak Convergence Approach to the Theory of Large Deviations. New York: Wiley.
   Google Scholar
• Budhiraja, A., and Dupuis, P. 2000. A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 20:39–61.
   Google Scholar
• Dembo, A., and Zeitouni, O. 1993. Large Deviations Techniques and Applications. Boston: Jones and Bartlett.
   Google Scholar
• Kato, T., and Ponce, G. 1988. Commutator estimayes and the Euler and Navier-Stokes equation. Commun. Pure Appl. Math. 41:891–907.
   Web of Science ®Google Scholar
• Li, Y. A., and Olver, P. J. 2000. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162:27–63.
   Web of Science ®Google Scholar
• Bona, J. L., and Smith, R. 1975. The initial value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. London A 278:555–601.
   Web of Science ®Google Scholar
• Da Prato, G., and Zabczyk, J. 1992. Stochastic Equations in Infinite Dimensions. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Stroock, D. W. 1984. An Introduction to the Theory of Largr Deviations. Berlin: Springer-Verlag.
   Google Scholar
• Barlow, M. T., and Yor, M. 1982. Semi-martingale inequalities via the Garsia-Rudemich-Rumsey lemma, and applications to local time. J Funct Anal 49:198–229.
   Web of Science ®Google Scholar
• Davis, B. 1976. On the Lp-norm of stochastic integrals and other martingales. Duke Math. J. 43:696–704.
   Google Scholar