The Annals of Probability

Rotation Numbers For Linear Stochastic Differential Equations

Ludwig Arnold and Peter Imkeller

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Let $dx=\sum_{i=0}^{m}A_ix\circ dW^i$ be a linear SDE in $\mathbb{R}^d$, generating the flow $\Phi_t$ of linear isomorphisms. The multiplicative ergodic theorem asserts that every vector $v\in\mathbb{R}^d\backslash\{0\}$ possesses a Lyapunov exponent (exponential growth rate) $\lambda(v)$ under $\Phi_t$, which is a random variable taking its values from a finite list of canonical exponents $\lambda_i$ realized in the invariant Oseledets spaces $E_i$. We prove that, in the case of simple Lyapunov spectrum, every 2-plane $p$ in $\mathbb{R}^d$ possesses a rotation number $\rho(p)$ under $\Phi_t$ which is defined as the linear growth rate of the cumulative inffinitesimal rotations of a vector $v_t$ inside $\Phi_t(p)$. Again, $\rho(p)$ is a random variable taking its values from a finite list of canonical rotation numbers $\rho_{ij}$ realized in span $(E_i, E_j)$. We give rather explicit Furstenberg-Khasminski-type formulas for the $\rho_{i,j}$. This carries over results of Arnold and San Martin from random to stochastic differential equations, which is made possible by utilizing anticipative calculus.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 130-149.

First available in Project Euclid: 29 May 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 93E03: Stochastic systems, general
Secondary: 34D08: Characteristic and Lyapunov exponents 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Stochastic differential equation random dynamical system cocycle stochastic flow anticipative calculus multiplicative ergodic theorem rotation number Furstenberg-Khasminskii formula


Arnold, Ludwig; Imkeller, Peter. Rotation Numbers For Linear Stochastic Differential Equations. Ann. Probab. 27 (1999), no. 1, 130--149. doi:10.1214/aop/1022677256.

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