The Annals of Probability

Rotation Numbers For Linear Stochastic Differential Equations

Ludwig Arnold and Peter Imkeller

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Abstract

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

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

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677256

Digital Object Identifier
doi:10.1214/aop/1022677256

Mathematical Reviews number (MathSciNet)
MR1681134

Zentralblatt MATH identifier
0944.60065

Subjects
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]

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

Citation

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


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