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Stochastic versions of the LaSalle theorem. (English) Zbl 0921.34057
The famous LaSalle theorem [see J. R. LaSalle, J. Differ. Equations 4, 57-65 (1968; Zbl 0159.12002)] for locating limit sets of nonautonomous systems is generalized to the case of ordinary stochastic differential equations driven by \(m\)-dimensional Brownian motion under the hypothesis of local Lipschitz continuity and at most linear-polynomial growth of drift and diffusion parts.
The proofs are carried out by a lemma due to R. Sh. Liptser and A. N. Shiryaev [ Martingale theory. Vyp. 38. Moskva: ‘Nauka’. (1986; Zbl 0654.60035)] on asymptotics of semimartingales, the well-known Kolmogorov-Centsov continuity theorem [I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. New York etc.: Springer-Verlag. (1991; Zbl 0734.60060; 1988; Zbl 0638.60065)], Burkholder type estimates and the uniform continuity of stochastic \(L^p\)-integrable martingales, and of course, by the help of Ito’s formula.
Thus, attracting deterministic sets, an estimation of exponential and polynomial growth rates can be found for stochastic differential equations. A series of examples illustrates the immense power of stochastic versions of LaSalle’s theorem and striking importance of that author’s work.

MSC:
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G35 Signal detection and filtering (aspects of stochastic processes)
93E15 Stochastic stability in control theory
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References:
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