Rotation numbers for linear stochastic differential equations
成果类型：
Article
署名作者：
Arnold, L; Imkeller, P
署名单位：
University of Bremen; Humboldt University of Berlin
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1999
页码：
130-149
关键词：
摘要：
Let dx = Sigma(i=0)(m) A(i)x circle dW(i) be a linear SDE in R-d, generating the flow Phi(t), of Linear isomorphisms. The multiplicative ergodic theorem asserts that every vector v is an element of R-d \ {0} possesses a Lyapunov exponent (exponential gran th 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 R-d possesses a rotation number rho(p) under Phi(t) which is defined as the linear growth rate of the cumulative infinitesimal rotations of a vector u(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-Khasminskii-type formulas for the rho(ij). This carries over results of Arnold and San Martin from random to stochastic differential equations, which is made possible by utilizing anticipative calculus.