Electronic Journal of Probability

Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations

Gerald Kager and Michael Scheutzow

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Let $Z$ be an $R^m$-valued semimartingale with stationary increments which is realized as a helix over a filtered metric dynamical system $S$. Consider a stochastic differential equation with Lipschitz coefficients which is driven by $Z$. We show that its solution semiflow $\phi$ has a version for which $\varphi(t,\omega)=\phi(0,t,\omega)$ is a cocycle and therefore ($S$,$\varphi$) is a random dynamical system. Our results generalize previous results which required $Z$ to be continuous. We also address the case of local Lipschitz coefficients with possible blow-up in finite time. Our abstract perfection theorems are designed to cover also potential applications to infinite dimensional equations.

Article information

Electron. J. Probab., Volume 2 (1997), paper no. 8, 17 pp.

Accepted: 2 December 1997
First available in Project Euclid: 26 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 28D10: One-parameter continuous families of measure-preserving transformations 34C35

stochastic differential equation random dynamical system cocycle perfection

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Kager, Gerald; Scheutzow, Michael. Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations. Electron. J. Probab. 2 (1997), paper no. 8, 17 pp. doi:10.1214/EJP.v2-22. https://projecteuclid.org/euclid.ejp/1453839984

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