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


Zbl 0684.34059
Full Text: DOI


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