Moment attractivity, stability and contractivity exponents of stochastic dynamical systems

Discrete and Continuous Dynamical Systems

Volumn 7, Issue 3, 2001, Pages 487-515

  Schurz, Henri ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Asymptotic behavior; Characteristic exponents; Moment attractivity exponents; Moment contractivity exponents; Moment Lyapunov exponents; Moment stability exponents; Sample Lyapunov exponents; Stochastic dynamical systems

Indexed keywords

EID: 0035589881     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2001.7.487     Document Type: Article

References (39)

1. Arnold, L. (1974): Stochastic differential equations: Theory and applications. Wiley
2. Arnold, L. (1984): A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44, 793-802
3. Arnold, L. and Wihstutz, V. (1986): Lyapunov exponents. Lecture Notes in Mathematics 1186, Springer
4. Arnold, L., Crauel, H. and Eckmann, J.-P. (1991): Lyapunov exponents. Lecture Notes in Mathematics 1486, Springer
5. Arnold, L. and Imkeller, P. (1995): Furstenberg-Khasminskii formulas for Lyapunov exponents via anticipative calculus. Stochastics Stochastics Rep. 54 (1-2), 127-168
6. Arnold, L. (1998): Random dynamical systems. Monographs in Mathematics, Springer
7. Baxendale, P.H. (1986): Asymptotic behavior of stochastic flows of diffeomorphisms. Lecture Notes in Mathematics 1203, 1-19
8. Baxendale, P.H. (1987): Moment stability and large deviations for linear stochastic differential equations. Probabilistic methods in mathematical physics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1985, 31-54
9. Baxendale, P.H. (1989): Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Theory Relat. Fields 81 (4), 521-554
10. Crauel, H., Debussche, A. and Flandoli, F. (1997): Random attractors. J. Dyn. Differ. Equations 9 (2), 307-341
11. Dynkin, E.B. (1965): Markov processes I, II. Die Grundlehren der mathematischen Wissenschaften 121/122. Springer
12. Freidlin, M.I. and Wentzell, A.D. (1998): Random perturbations of dynamical systems (second edition). Grundlehren der mathematischen Wissenschaften 260, Springer
13. Friedman, A. (1975): Stochastic differential equations and applications I, II. Academic Press
14. Gard, T.C. (1988): Introduction to stochastic differential equations. Marcel Dekker
15. Hale, J.K. (1988): Asymptotic behavior of dissipative systems. Mathematical Surveys and Monographs 25, Providence, RI: American Mathematical Society
16. Imkeller, P. and Scheutzow, M. (1999): On the spatial asymptotic behavior of stochastic flows in Euclidean space. Ann. Probab. 27 (1), 109-129
17. Itô, K. (1944): Stochastic integral. Proc. Imp. Acad. Tokyo 20, 519-524
18. Itô, K. (1951a): On a formula concerning stochastic differentials. Nagoya Math. J. 1, 55-65
19. Itô, K. (1951b): On stochastic differential equations. Amer. Math. Soc. Memoirs 4, 1-51
20. Kannan, D. (1979): An introduction to stochastic processes, North Holland
21. Khas'minskij, R.Z. (1980): Stochastic stability of differential equations. Sijthoff & Noordhoff
22. Kifer, Y. (1988): Random perturbation of dynamical systems. Birkhäuser
23. Kliemann, W. and Sri Namachchivaya, N. (1995): Nonlinear dynamics and stochastic mechanics. Modelling Series 5, CRC Press
24. Kloeden, P.E., Platen, E. and Schurz, H. (1997): Numerical solution of stochastic differential equations through computer experiments (2nd edition). Universitext, Springer
25. Krylov, N.V. (1990): A simple proof of the existence of a solution of Ito's equation with monotone coefficients. Theory Probab. Appl. 35 (3), 583-587
26. Krylov, N.V. (1995): Introduction to the theory of diffusion processes. Translations of Mathematical Monographs 142, Providence, RI: American Mathematical Society
27. Mao, X. (1994): Exponential stability of stochastic differential equations. Pure and Applied Mathematics 182, Marcel Dekker
28. Mao, X. (1999): Stochastic versions of the LaSalle theorem. J. Differential Equations 153, 175-195
29. Mil'shtein, G.N. (1988): Numerical integration of stochastic differential equations (in Russian). Uralski University Press, Sverdlovsk (English translation by Kluwer, Dordrecht, 1995)
30. Oseledec, V.I. (1968): A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow. Math. Soc. 19, 197-231
31. Protter, P. (1990): Stochastic integration and differential equations, Springer
32. Schurz, H. (1997): Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications. Logos-Verlag (also Report 11, WIAS, Berlin, 1996)
33. Schurz, H. (1999): On moment-dissipative stochastic dynamical systems, Dyn. Syst. Applic. (accepted), Technical Report 214, University of Kaiserslautern, July 1999
34. Schurz, H. (1999): Linear-and partial-implicit numerical methods for nonlinear stochastic differential equations. Manuscript, University of Minnesota, Minneapolis
35. Schurz, H. (1999): General principles for numerical approximation of stochastic processes on stochastically weak Banach spaces. Manuscript. IMA Report 1669, University of Minnesota, Minneapolis, December 1999
36. Schurz, H. (1999): A brief introduction to numerical methods for (ordinary) stochastic differential equations. Invited chapter for the Handbook of Stochastic Analysis. Manuscript. IMA Report 1670, University of Minnesota, Minneapolis, December 1999
37. Shiryaev, A.N. (1996): Probability (2nd edition). Graduate Texts in Mathematics 95, Springer
38. Talay, D. (1995): Simulation of stochastic differential systems. Probabilistic methods in applied physics (Kree, P. & Wedig, W., eds.), 54-96. Lecture Notes in Physics 451, Springer
39. Wagner, W. and Platen, E. (1978): Approximation of Itô integral equations. Preprint ZIMM, Acad. Sci. GDR