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P-th moment growth bounds of infinite-dimensional stochastic evolution equations

Published online by Cambridge University Press:  14 November 2011

Kai Liu
Kai Liu
Affiliation:
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow Gl 1XH, Scotland, U.K. e-mail: kai@stams.strath.ac.uk
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Extract

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 107 - 121
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Arnold, V.Stochastic differential equations: theory and application (New York: Wiley, 1974).Google Scholar
2Curtain, R. and Pritchad, A.. Infinite dimensional linear system theory, Lecture Notes in Control and Inform. Sci. 8 (New York: Springer, 1978).CrossRefGoogle Scholar
3Khas'minskii, R. and Mandrekar, V.. On the stability of solutions of stochastic evolution equations. In The Dynkin Festschrift, Progress in Probability 34, 185–97 (Boston, MA: Birkhauser, 1994).CrossRefGoogle Scholar
4Ladde, G.. Differential inequalities and stochastic functional differential equations. J. Math. Phys. 15 (1974), 738–43.CrossRefGoogle Scholar
5Liu, K.. Stability with a general rate function for a class of stochastic evolution equations in infinite dimensional spaces. J. Austral. Math. Soc. (Series A) 63 (1997), 128–44.CrossRefGoogle Scholar
6Liu, K.. On stability for a class of semilinear stochastic evolution equations. Sioch. Proc. Appl. 70 (1997), 219–41.CrossRefGoogle Scholar
7Mao, X.. Exponential stability in mean square for stochastic differential equations. Stochastic Anal. Appl. 8(1990), 91–103.CrossRefGoogle Scholar
8Mao, X.. Stability of stochastic differential equations with respect to semimartingales. Pitman Research Notes in Mathematics Series 251 (Longman Scientific and Technical, 1991).Google Scholar
9Mao, X.. Almost sure asymptotic bounds for a class of stochastic differential equations. Stochastics 41 (1992), 57–69.Google Scholar
10Mao, X.. Exponential stability of stochastic differential equations (New York: Marcel Dekker, 1994).Google Scholar
11Mao, X.. p-th moment exponential stability of large-scale stochastic delay systems in hierarchical form. Differential Equations Control Theory (to appear).Google Scholar
12Mao, X.. Exponential stability in mean square of neutral stochastic differential functional equations. Systems Control Lett, (to appear).Google Scholar
13Mao, X. and Markus, L.. Energy bounds for nonlinear dissipative stochastic differential equations with respect to semimartingales. Stochastics 37 (1991), 1–14.Google Scholar
14Métivier, M.. Semimartingales (Berlin: Walter de Gruyter, 1982).CrossRefGoogle Scholar
15Mizel, V. and Trutzer, V.. Stochastic hereditary equations: existence and asymptotic stability. J. Integral Equations 7 (1984), 1–72.Google Scholar
16Pardous, E.. Équations aux dérivées partielles stochastiques non linéaires monotones (Thesis, Université Paris XI, 1975).Google Scholar
17Prato, G. da and Zabczyk, J.. Stochastic equations in infinite dimensions (Cambridge: Cambridge University Press, 1992).CrossRefGoogle Scholar
18Viot, M.. Solutions faibles d équations aux dérivées partielles stochastique non linéaires (Thesis, Université Paris XI, 1978).Google Scholar