Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-12T05:44:15.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability with a general rate function for a class of stochastic evolution equations in infinite dimensional spaces

Part of: Stochastic analysis Stochastic systems and control

Published online by Cambridge University Press:  09 April 2009

Kai Liu
Kai Liu
Affiliation:
Department of Statistics and Modelling Science University of StrathclydeGlasgow G1 1XH, ScotlandU. K. e-mail address: kai@stams.strath.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is to investigate the almost sure stability with a certain rate function λ(t) for a class of stochastic evolution equations in infinite dimensional spaces under various sufficient conditions. The results obtained here include exponential and polynomial stability as special cases. Much more refined sufficient conditions than the usual ones, for example, those described in [14], are obtained under our framework by the method of Liapunov functions. Two examples are given to illustrate our theory.

MSC classification

Secondary: 93E03: Stochastic systems, general 60H10: Stochastic ordinary differential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 1 , August 1997 , pp. 128 - 144
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Arnold, L., Stochastic differential equation: theory and applications (Wiley, New York, 1974).Google Scholar
[2]Arnold, L., ‘A formula connecting sample and moment stability of linear stochastic systems’, SIAM J. Appl. Math. 44 (1984), 793–802.CrossRefGoogle Scholar
[3]Caraballo, T., ‘Asymptotic exponential stability of stochastic partial differential equations with delay’, Stochastics 33 (1990), 27–47.Google Scholar
[4]Carverhill, A., ‘Flows of stochastic dynamical systems: ergodic’, Stochastics 14 (1985), 273–317.CrossRefGoogle Scholar
[5]Chappell, M., ‘Bounds for average Liapunov exponents of gradient stochastic systems,’ in: Lecture Notes in Math. 1186 (Springer, Berlin, 1984), pp. 292–307.Google Scholar
[6]Chow, P. L., ‘Function-space differential equations associated with a stochastic partial differential equation’, Indiana Univ. Math. J. 25 (1976), 609–627.CrossRefGoogle Scholar
[7]Chow, P. L., ‘Stability of nonlinear stochastic evolution equations’, J. Math. Anal. Appl. 89 (1982), 400–419.CrossRefGoogle Scholar
[8]Crauel, H., ‘Liapunov numbers of Markov solutions of linear stochastic systems’, Stochastics 14 (1984), 11–28.CrossRefGoogle Scholar
[9]Curtain, R. (ed), ‘Stability of stochastic dynamical systems’, in: Lecture Notes in Math. 294 (Springer, Berlin, 1972).Google Scholar
[10]Curtain, R., ‘Stability of stochastic partial differential equation’, J. Math. Anal. Appl. 79 (1981), 352–369.CrossRefGoogle Scholar
[11]Has'minskii, R. Z., ‘Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems’, Theory Probab. Appl. 12 (1969), 144–147.CrossRefGoogle Scholar
[12]Haussmann, U. G., ‘Asymptotic stability of the linear Itô equation in infinite dimensional’, J. Math. Anal. Appl. 65 (1978), 219–235.CrossRefGoogle Scholar
[13]Mao, X., ‘Almost sure polynomial stability for a class of stochastic differential equations’, Quarterly J. Math. Oxford (2). 43 (1992), 339–348.CrossRefGoogle Scholar
[14]Mao, X., Stability of stochastic differential equations with respect to semimartingales, Pitman Res. Notes Math. Ser. 251 (Longman Sci. Tech., Harlow, 1991).Google Scholar
[15]Mao, X., Exponential stability of stochastic differential equations (Marcel Dekker, New York, 1994).Google Scholar
[16]Mao, X., ‘Exponential stability for nonlinear stochastic differential equations with respect to semimartingale’, Stochastics 28 (1989), 343–355.Google Scholar
[17]Métivier, M., Semimartingales (de Gruyter, Berlin, 1982).CrossRefGoogle Scholar
[18]Pardoux, E., Équations aux dérivées partielles stochastiques non linéaires monotones (Thesis, Université Paris XI, 1975).Google Scholar
[19]Da Prato, G. and Zabczyk, J., Stochastic equations in infinite dimensions (Cambridge Univ. Press., Cambridge, 1992).CrossRefGoogle Scholar
[20]Viot, M., Solutions faibles d'équations aux dérivées partielles stochastique non linéaires (Thesis, Université Paris VI, 1978).Google Scholar