Furstenberg-khasminskii formulas for lyapunov exponents via anticipative calculus

Abstract

According to a classical formula due to Furstenberg and Khasminskii the top Lyapunov exponent of a random linear cocycle can be expressed as a phase average of a function over projective space. Using a description of the Oseledets spaces as intersections of two flags of random subspaces of the phase space, one forward and one backward in time, we derive similar formulas for the remaining Lyapunov exponents of the cocycle generated by a linear stochastic differential equation. This approach makes elements of anticipative calculus enter the scene quite naturally. Indeed, in the correction terms appearing in the analogues of the Furstenberg-Khasminskii formula, averages of the Malliavin gradient of the orthogonal projectors on the Oseledets spaces come into play explicitly