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Lyapunov exponent estimates of a class of higher-order stochastic Anderson models

Authors: Dan Tang and Lijun Bo
Journal: Proc. Amer. Math. Soc. 136 (2008), 4033-4043
MSC (2000): Primary 60H15, 34A34, 49N60
Published electronically: June 5, 2008
MathSciNet review: 2425745
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Abstract: In this article, we propose a class of high-order stochastic partial differential equations (SPDEs) for spatial dimensions $ d\leq5$ which might be called high-order stochastic Anderson models. This class of the equations is perturbed by a space-time white noise when $ d\leq 3$ and by a space-correlated Gaussian noise when $ d=4,5$. The objectives of this article are to get some estimates on the Lyapunov exponent of the solutions and to study the convergence rates of the chaos expansions of the solutions for the models.

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  • 1. L. Bo, Y. Wang, Stochastic Cahn-Hilliard partial differential equations with Lévy spacetime noises, Stoch. Dyn. 6 (2006), 229-244. MR 2239091 (2007i:60074)
  • 2. L. Bo, Y. Wang, and L. Yan, Higher-order Stochastic partial differential equations with branching noises, Front. Math. China 3 (2008), 15-35. MR 2373092.
  • 3. C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution and of its density, Bernoulli 7 (2001), 777-816. MR 1867082 (2002i:60109)
  • 4. C. Cardon-Weber and A. Millet, On strongly Petrovskiĭ's parabolic SPDEs in arbitrary dimension and application to the stochastic Cahn-Hilliard equation, J. Theory Prob. 17 (2004), 1-49. MR 2054575 (2005f:60136)
  • 5. R. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDEs, Electron. J. Prob. 4 (1999), 1-29. MR 1684157 (2000b:60132)
  • 6. Y. Hu, Chaos expansion of heat equations with white noise potentials, Potential Anal. 16 (2002), 45-66. MR 1880347 (2002k:60133)
  • 7. C. Mueller, Long time existence for the heat equation with a noise term, Prob. Th. Rel. Fields 90 (1991), 505-517. MR 1135557 (93e:60120)
  • 8. C. Mueller and R. Tribe, A measure-valued process related to the parabolic Anderson model, Progress in Prob. 52, Birkhäuser, Basel (2002), 219-227. MR 1958819
  • 9. D. Nualart and B. Rozovskii, Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise, J. Funct. Anal. 149 (1997), 200-225. MR 1471105 (98m:60100)
  • 10. D. Nualart and M. Zakai, Generalized Brownian functionals and the solution to a stochastic partial differential equation, J. Funct. Anal. 84 (1989), 279-296. MR 1001461 (90m:60076)
  • 11. H. Uemura, Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behavior, Stoch. Anal. Appl. 14 (1996), 487-506. MR 1402691 (97e:60109)
  • 12. J. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math. 1180, Springer, Berlin (1986), 265-439. MR 876085 (88a:60114)

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Additional Information

Dan Tang
Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China

Lijun Bo
Affiliation: Department of Mathematics, Xidian University, Xi’an 710071, People’s Republic of China

Keywords: High-order SPDEs, stochastic Anderson models, Lyapunov exponent, convergence rates.
Received by editor(s): October 18, 2007
Published electronically: June 5, 2008
Additional Notes: This work was supported by the LPMC at Nankai University and the NSF of China (No. 10471003).
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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