Rotation numbers for linear stochastic differential equations. (English) Zbl 0944.60065

Arnold, Ludwig; Imkeller, Peter

Ann. Probab. 27, No. 1, 130-149 (1999).

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

Keywords:

Stratonovich stochastic differential equation; random dynamical system; cocycle; stochastic flow; Malliavin calculus; multiplicative ergodic theorem; Furstenberg-Khasminskij formula

Citations:

Zbl 0684.34059

References:

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