Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations
Michael Scheutzow (Technische Universität Berlin)
Abstract
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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Pages: 1-17
Publication Date: December 2, 1997
DOI: 10.1214/EJP.v2-22
References
- L. Arnold, Random dynamical systems, Springer, Berlin (to appear) (1998). Math Review article not available.
- L. Arnold and M. Scheutzow, Perfect cocycles through stochastic differential equations, Probab. Theory Relat. Fields 101, (1995) 65-88. Math Review link
- D.L. Cohn, Measure theory, Birkh"auser, Boston (1980). Math Review link
- C. Dellacherie and P.A. Meyer, Probabilities and potential, North Holland, Amsterdam (1978). Math Review link
- J. Dugundji, Topology, Allyn and Bacon, Boston (1966). Math Review link
- R. Getoor, Excessive measures, Birkh"auser, Boston (1990). Math Review link
- G. Kager, Zur Perfektionierung nicht invertierbarer grober Kozykel, Ph.D. thesis, Technische Universit"at Berlin. Math Review article not available.
- H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d''Et'e de Prob. de Saint Flour XII. Lecture Notes in Mathematics 1097, 143-303. Springer, Berlin (1984). Math Review link
- H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge (1990). Math Review link
- P.A. Meyer, La perfection en probabilit'e. S'eminaire de Probabilit'e VI. Lecture Notes in Mathematics 258, 243-253, Springer, Berlin (1972). Math Review link
- P.A. Meyer, Flot d'une 'equation diff'erentielle stochastique. S'eminaire de Probabilit'e XV}. Lecture Notes in Mathematics 850, 103-117, Springer, Berlin (1981). Math Review link
- S.E.A. Mohammed and M. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part I: the multiplicative ergodic theory, Ann. Inst. Henri Poincar'e (Prob. et Stat.) 32, (1996) 69-105. Math Review link
- P. Protter, Semimartingales and measure preserving flows, Ann. Inst. Henri Poincar'e (Prob. et Stat.) 22, (1986) 127-147. Math Review link
- P. Protter, Stochastic integration and differential equations, Springer, Berlin (1992). Math Review link
- M. Scheutzow, On the perfection of crude cocycles, Random and Comp. Dynamics, 4, (1996) 235-255. Math Review article not available.
- M. Sharpe, General theory of Markov processes, Academic Press, Boston (1988). Math Review link
- J.B. Walsh, The perfection of multiplicative functionals, S'eminaire de Probabilit'e VI. Lecture Notes in Mathematics 258, 233-242. Springer, Berlin (1972). Math Review link
- R. Zimmer, Ergodic theory and semisimple groups, Birkh"auser, Boston (1984). Math Review link

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