Abstract
This paper discusses the ergodicity of a linear stochastic partial differential equation driven by Lévy noise.
Similar content being viewed by others
References
D. Barbu and B. Gheorghe, Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients, Czechoslovak Math. J., 2002, 52(127): 87–95.
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.
G. Da Prato, A. Debussche, and G. Beniamin, Some properties of invariant measures of non symmetric disspative stochstic systems, Probab. Theory Related Fields, 2002, 123: 355–380.
Y. Liu and H. Z. Zhao, Reprensentation of pathwise staionary solutions of stochastic Burgers’ equations, Stochastics and Dynamics, 2009, 9(4): 613–634.
R. Mikulevicius, H. Pragarauskas, On Cauchy-Dirichlet problem for parabolic quasilinear SPDEs, Potential Analysis, 2006, 25: 37–75.
R. Mikulevicius, H. Pragarauskas, and N. Sonnadara, On the Cauchy-Dirichlet problem in the half space for parabolic SPDEs in weighted Hoelder spaces, Acta Appl. Math., 2007, 97: 129–149.
S. E. A. Mohammed, T. S. Zhang, and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differenial equations, Memoirs of the American Mathematical Society, 2006.
Y. Xie, Existence and uniqueness of solutions of SPDE with non-Lipschitz and non-time-homogeneous coefficients, J. Xuzhou Norm. Univ. Nat. Sci. Ed., 2007, 25(1): 1–5.
S. Albeverio, J. L. Wu, and T. S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 1998, 74(1): 21–36.
D. Applebauam and J. L. Wu, Stochastic partial differential equations driven by Lévy space-time white noise, Random Oper. Stochastic Equations, 2000, 8(3): 245–259.
Z. Dong, The uniqueness of invariant measure of the Burgers equation driven by Lévy processes, Journal of Theoretical Probability, 2008, 21(2): 322–335.
Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy Noise, Science in China Series A: Mathematics, 2009, 52(7): 1–29.
Z. Dong and T.G. Xu, One-dimensional stochastic burgers equation driven by Lévy processes, Journal Functional Analysis, 2007, 243: 631–678.
E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results, Probab. Theory Related Fields, 2007, 137: 161–200.
Erwan Saint Loubert Bié, Étude d’une EDPS conduite par un bruit poissonnien, Probability Theory and Related Fields, 1998, 111(2): 287–321.
A. Truman and J. L. Wu, On a stochastic nonlinear eqnation arising from 1D integro-differential scalar conservation laws, Journal of Functional Analysis, 2006, 238: 612–635.
A. Truman and J. L. Wu, Stochastic Burgers equation with Lévy space-time white noise, Probabilistic Methods in Fluids, proceedings of the Swansea 2002 workshop.
Sandra Cerrai, Second Order PDE’s in Finite and Infinite Demension, springer-Verlag Berlin Heidelberg, 2001.
J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001.
F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 1995, 171: 119–141.
B. Ferrario, Ergodic results for stochastic Navier-Stokes equation, Stochastics and Stochastics Reports, 1997, 60: 271–288.
G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Birkhäuser Verlag, 2004.
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was supported by the Key Laboratory of Random Complex Structures and Data Scienc, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 973 Project (2006CB8059000), Science Fund for Creative Research Groups (10721101), the National Science Foundation of China (10671197), and the Science Foundation of Jiangsu Province (BK2006032, 06-A-038, 07-333).
Rights and permissions
About this article
Cite this article
Dong, Z., Xie, Y. Ergodicity of linear SPDE driven by Lévy noise. J Syst Sci Complex 23, 137–152 (2010). https://doi.org/10.1007/s11424-010-9269-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-010-9269-0