Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise : Itô's case

Stochastic Analysis and Applications

Volumn 18, Issue 4, 2000, Pages 581-615

  Chueshov, Igor D ^a   Vuillermot, Pierre A ^b  

^a V N KARAZIN KHARKIV NATIONAL UNIVERSITY   (Ukraine)

^b UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034342828     PISSN: 07362994     EISSN: None     Source Type: Journal    
DOI: 10.1080/07362990008809687     Document Type: Article

References (53)

1. Arnold, L. : Stochastic Differential Equations : Theory and Applications. London New York Sydney Toronto : John Wiley and Sons, 1973.
2. Arnold, L. : Zufällige Dynamische Systeme. Jahresbericht der Deutschen Mathematiker-Vereinigung 96, 3, 1994, pp. 85-100.
3. Arnold, L., Chueshov, I.D. : Order-Preserving Dynamical Systems : Equilibria, Attractors, Applications. Dynamics and Stability of Systems 13, 3, 1998, pp. 265-280.
4. Arnold, L., Crauel, H. : Random Dynamical Systems. In : Arnold, L., Crauel, H., Eckmann, J.P. (eds.) : Lyapunov Exponents, Oberwohlfach 1990 (Lect. Notes Math., vol. 1486, pp. 1-22) Berlin Heidelberg New York : Springer, 1991.
5. Arnold, L., Wihstutz, V. : Lyapunov Exponents : A Survey. In : Arnold, L., Wihstutz, V. (eds.) : Lyapunov Exponents, Bremen 1984 (Lect. Notes Math., vol. 1186, pp. 1-84) Berlin Heidelberg New York : Springer, 1986.
6. Bergé, B., Chueshov, I.D. and Vuillermot, P.A. : Lyapunov Exponents and Stability for Nonlinear SPDE's Driven by Finite-Dimensional Wiener Processes. C.R. Acad. Sci. Paris 329, 1, 1999, pp. 215-220.
7. Bernfeld, S.R., Hu, Y.Y., Vuillermot, P.A. : Large-Time Asymptotic Equivalence for a Class of Non-Autonomous Semilinear Parabolic Equations. Bull. Sci. Math. 122, 5, 1998, pp. 337-368.
8. Brézis, H. : Analyse Fonctionnelle : Théorie et Applications. Paris New York : Masson, 1983.
9. Carr, J. : Applications of Centre Manifold Theory. (Applied Mathematical Sciences, vol. 35) Berlin Heidelberg New York : Springer, 1981.
10. Chow, S.N., Hale, J.K. : Methods of Bifurcation Theory. (Grundlehren der mathematischen Wissenschaften, vol. 251) Berlin Heidelberg New York : Springer, 1982.
11. Chueshov, I.D. : On Approximate Inertial Manifolds for Stochastic Navier-Stokes Equations. J. of Math. Anal. and Appl. 196, 1, 1995, pp. 221-236.
12. Chueshov, I.D., Vuillermot, P.A. : On the Large-Time Dynamics of a Class of Random Parabolic Equations. C.R. Acad. Sci. Paris 322. I, 1996, pp. 1181-1186.
13. Chueshov, I.D., Vuillermot, P.A. : On the Large-Time Dynamics of a Class of Parabolic Equations subjected to Homogeneous White Noise : Stratonovitch's Case. C.R. Acad. Sci. Paris 323, I, 1996, pp. 29-33.
14. Chueshov, I.D., Vuillermot, P.A. : Long-Time Behavior of Solutions to a Class of Quasilinear Parabolic Equations with Random Coefficients. Ann. Inst. Henri-Poincaré, Nonlinear Analysis 15, 2, 1998, pp. 191-232.
15. Chueshov, I.D., Vuillermot, P.A. : Long-Time Behavior of Solutions to a Class of Stochastic Parabolic Equations with Homogeneous White Noise : Stratonovitch's Case. Probab. Theory Relat. Fields 112, 2,1998, pp. 149-202.
16. Chueshov, I.D., Vuillermot, P.A. : On the Large-Time Dynamics of a Class of Parabolic Equations subjected to Homogeneous White Noise : Itô's Case. C.R. Acad. Sci. Paris 326, I, 1998, pp. 1299-1304.
17. Chueshov, I.D., Vuillermot, P.A. : Monotone Methods for the Qualitative Analysis of Nonlinear Random and Stochastic Partial Differential Equations. Book in preparation.
18. Crauel, H., Debussche, A., Flandoli, F. : Random Attractors. J. of Dynamics and Differential Equations 9, 2, 1997, pp. 307-341.
19. Crauel, H., Flandoli, F. : Attractors for Random Dynamical Systems. Probab. Theory Relat. Fields 100, 1994, pp. 365-393.
20. Da Prato, G., Debussche, A., Temam, R. : Stochastic Burger's Equation. Nonlin. Diff. Eq. Appl. 1, 1994, pp. 389-402.
21. Da Prato, G., Zabczyk, J. : Stochastic Equations in Infinite Dimensions. (Encyclopedia of Mathematics and its Applications, vol. 44) : Cambridge University Press, 1992.
22. Da Prato, G., Zabczyk, J. : Ergodicity for Infinite-Dimensional Systems. (London Mathematical Society Lecture Note Series, vol. 229) : London Mathematical Society, 1996.
23. Doob, J.L. : Stochastic Processes. London New York Sydney Toronto : John Wiley and Sons, 1953.
24. Flandoli, F. : Stochastic Flow and Lyapunov Exponents for Abstract Stochastic PDE's of Parabolic Type. In : Proceedings Conf. on Lyapunov Exponents, Oberwohlfach 1990, Arnold, L., Crauel, H., Eckmann, J.P. (eds.) (Lect. Notes Math., vol. 1486) Berlin Heidelberg New York : Springer, 1991.
25. Flandoli, F. : Dissipativity and Invariant Measures for Stochastic Navier-Stokes Equations. Nonlin. Diff. Eq. Appl. 1, 1994, pp. 403-423.
26. Flandoli, F., Gatarek, D. : Martingale and Stationary Solutions for Stochastic Navier-Stokes Equations. Probab. Theory Relat. Fields 102, 3, 1995, pp. 367-391.
27. Flandoli, F., Maslowski, B. : Ergodicity of the 2-D Navier-Stokes Equation under Random Perturbations. Commun. Math. Phys. 172, 1, 1995, pp. 119-141.
28. Friedman, A. : Partial Differential Equations. Krieger Publishing Co., 1976.
29. Gagliardo, E. : (Ulteriori) Proprietà di Alcune Classi di Funzioni in più Variabili. Ricerche Mat. 7, 102-137 (1958) and Ricerche Mat. 8, 1959, pp. 24-51.
30. Gagliardo, E. : (Ulteriori) Proprietà di Alcune Classi di Funzioni in più Variabili. Ricerche Mat. 7, 102-137 (1958) and Ricerche Mat. 8, 1959, pp. 24-51.
31. Gihman, I.I., Skorohod, A.V. : Stochastic Differential Equations. (Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 72) Berlin Heidelberg New York : Springer, 1972.
32. Girya, T.V., Chueshov I.D. : Inertial Manifolds and Stationary Measures for Stochastically Perturbed Dissipative Dynamical Systems. Math. Sbornik 186, 1, 1995, pp. 29-46.
33. Grecksch, W., Schmalfuss, B. : Approximation of the Stochastic Navier-Stokes Equation. Math. Appl. Comput. 15, 3, 1996, pp. 227-239.
34. Has'minskii, R.Z. : Stochastic Stability of Differential Equations. Sijthoff and Nordhof : Alphen, 1980.
35. Hess, P. : Periodic-Parabolic Boundary-Value Problems and Positivity. (Pitman Research Notes in Mathematics, vol. 247) Harlaw, Langman Sci. Tech., 1991.
36. Icbikawa, A. : Semilinear Stochastic Evolution Equations : Boundedness, Stability and Invariant Measures. Stochastics 12, 1984, pp. 1-39.
37. Ikeda, N., Watanabe, S. : Stochastic Differential Equations and Diffusion Processes. (North-Holland Mathematical Library, vol. 24) Amsterdam Oxford New York : North-Holland Publishing Company, 1981.
38. Itô, K., Mc Kean, H.P. : Diffusion Processes and their Sample Paths. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 125) Berlin Heidelberg New York : Springer, 1974.
39. Kushner, H.J. : Stochastic Stability and Control. (Mathematics in Science and Engineering, vol. 33) New York London : Academic Press, 1967.
40. Morimoto, H. : Attractors of Probability Measures for Semilinear Stochastic Evolution Equations. Stoch. Anal. Appl. 10, 1992, pp. 205-212.
41. Neças, J. : Introduction to the Theory of Nonlinear Elliptic Equations. London New York Sydney Toronto : John Wiley and Sons, 1986.
42. Nirenberg, L. : On Elliptic Partial Differential Equations. Ann. Scuola Norm. Sup. Pisa 13, 1959, pp. 115-162.
43. Revuz, D., Yor, M. : Continuous Martingales and Brownian Motion. (Grundlehren Math. Wiss., Bd. 293) Berlin Heidelberg New York : Springer, 1994.
44. Schmalfuss, B. : Measure Attractors and Stochastic Attractors. Institut für Dynamische Systeme Report 332, Universität Bremen, 1995.
45. Tuckwell, H.C. : The Effects of Random Selection on Gene Frequency. Math. Biosc. 30, 1976, pp. 113-128.
46. Turelli, J.M. : Random Environments and Stochastic Calculus. Theor. Popul. Biology 12, 1977, pp. 140-178.
47. Twardowska, K. : An Approximation Theorem of Wong-Zafcaï Type for Nonlinear Stochastic Partial Differential Equations. Stoch. Anal. Appl. 13, 5, 1995, pp. 601-626.
48. N, I. Global Stabilization Processes. J. Differential Equations 94, 2, 1991, pp. 228-253.
49. N, II. Codimension-One Stable Manifolds. Diff. Int. Eqs. 5, 3, 1992, pp. 693-720.
50. N, III. Center Curves and Liapounov Stability. Nonlinear Anal., TMA 22, 5, 1994, pp. 533-559.
51. N. Proc. Amer. Math. Soc. 116, 3, 1992, pp. 775-782.
52. Wong, E. : Stochastic Processes in Information and Dynamical Systems. (Mc Graw-Hill Series in, Systems Science) New York San Francisco : Mc Graw-Hill, 1971.
53. Wong, E., Zakaï, M. : Riemann-Stieltjes Approximations of Stochastic Integrals. Z. Wahrscheinlichkeitstheorie u. verw. Geb. 12, 1969, pp. 87-97.