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1997 Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations
Gerald Kager, Michael Scheutzow
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Electron. J. Probab. 2: 1-17 (1997). DOI: 10.1214/EJP.v2-22


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.


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Gerald Kager. Michael Scheutzow. "Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations." Electron. J. Probab. 2 1 - 17, 1997.


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

zbMATH: 0888.60044
MathSciNet: MR1485117
Digital Object Identifier: 10.1214/EJP.v2-22

Primary: 60H10
Secondary: 28D10 , 34C35

Keywords: Cocycle , perfection , Random dynamical system , Stochastic differential equation


Vol.2 • 1997
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