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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)

##### MSC:
 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
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##### References:
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