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January 1999 Rotation Numbers For Linear Stochastic Differential Equations
Ludwig Arnold, Peter Imkeller
Ann. Probab. 27(1): 130-149 (January 1999). DOI: 10.1214/aop/1022677256


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.


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Ludwig Arnold. Peter Imkeller. "Rotation Numbers For Linear Stochastic Differential Equations." Ann. Probab. 27 (1) 130 - 149, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0944.60065
MathSciNet: MR1681134
Digital Object Identifier: 10.1214/aop/1022677256

Primary: 60H10 , 93E03
Secondary: 34D08 , 34F05

Keywords: anticipative calculus , Cocycle , Furstenberg-Khasminskii formula , multiplicative ergodic theorem , Random dynamical system , rotation number , Stochastic differential equation , stochastic flow

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 1 • January 1999
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