Stochastic versions of the LaSalle theorem.

*(English)*Zbl 0921.34057The 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.

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.

Reviewer: Henri Schurz (Berlin)

##### 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 |

##### Keywords:

LaSalle invariance principle; stochastic stability; stochastic (ordinary) differential equations; stochastic Lyapunov method
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\textit{X. Mao}, J. Differ. Equations 153, No. 1, 175--195 (1999; Zbl 0921.34057)

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