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Rotation numbers for linear stochastic differential equations. (English) Zbl 0944.60065
Let \(\Phi\) denote the flow generated by the linear Stratonovich stochastic differential equation in \(\mathbb{R}^d\), \[ dX_t= A_0X_t dt+ \sum^m_{i=1} A_iX_t\circ dW^i_t,\quad t\geq 0, \] where \(A_0,\dots, A_m\in \mathbb{R}^{d\times d}\) and \(W= (W^1,\dots, W^m)\) is an \(m\)-dimensional standard Brownian motion. Under a sufficient condition for simple Lyapunov spectrum, the authors prove that every 2-plane \(p\) in \(\mathbb{R}^d\) possesses a rotation number \(\rho(p)\) under \(\Phi\). The rotation number \(\rho(p)\) is a random variable taking its values in a finite set of canonical rotation numbers for which an explicit Furstenberg-Khasminkij type formula is established. The paper translates former results of L. Arnold and L. San Martin [J. Dyn. Differ. Equations 1, No. 1, 95-119 (1989; Zbl 0684.34059)] on the multiplicative ergodic theorem for rotation numbers from the context of a random differential equation to that of a stochastic differential equation. The difficulty one meets here is that quantities coming from the multiplicative ergodic theorem can anticipate the driving Brownian motion \(W\); this explains why the Malliavin calculus plays a crucial role in the authors’ approach.
Reviewer: R.Buckdahn (Brest)

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
60H07 Stochastic calculus of variations and the Malliavin calculus
Full Text: DOI
[1] ARNOLD, L. 1995. Six lectures on random dynamical systems. Dynamical Systems. Lecture Notes in Math. 1609 1 43. Springer, Berlin.
[2] ARNOLD, L. 1998. Random Dynamical Systems. Springer, Berlin. · Zbl 0938.37031 · doi:10.1080/02681119808806264
[3] ARNOLD, L. and IMKELLER, P. 1995. Furstenberg Khasminskii formulas for Lyapunov exponents via anticipative calculus. Stochastics Stochastics Rep. 54 127 168. · Zbl 0857.60052
[4] ARNOLD, L. and IMKELLER, P. 1996. Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory. Stochastic Process. Appl. 62 19 54. · Zbl 0847.60037 · doi:10.1016/0304-4149(95)00081-X
[5] ARNOLD, L. and SAN MARTIN, L. 1989. A multiplicative ergodic theorem for rotation numbers. J. Dynam. Differential Equations 1 95 119. · Zbl 0684.34059 · doi:10.1007/BF01048792
[6] ARNOLD, L. and SCHEUTZOW, M. 1995. Perfect cocycles through stochastic differential equations. Probab. Theory Related Fields 101 65 88. · Zbl 0821.60061 · doi:10.1007/BF01192196
[7] IMKELLER, P. 1999. The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations. Potential Anal. · Zbl 0924.60031 · doi:10.1023/A:1008680717092
[8] KUNITA, H. 1990. Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. · Zbl 0743.60052
[9] NUALART, D. 1995. The Malliavin Calculus and Related Topics. Springer, Berlin. · Zbl 0837.60050
[10] OSELEDETS, V. I. 1968. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 197 231. · Zbl 0236.93034
[11] RUFFINO, P. 1997. Rotation numbers for stochastic dynamical systems. Stochastics Stochastics Rep. 60 289 318. · Zbl 0883.60062
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