THE METRIC ENTROPY OF RANDOM DYNAMICAL SYSTEMS IN A HILBERT SPACE: CHARACTERIZATION OF INVARIANT MEASURES SATISFYING PESIN'S ENTROPY FORMULA

摘要

Consider a random cocycle Φ on a separable infinite-dimensional Hilbert space preserving a probability measure μ, which is supported on a random compact set K. We show that if Φ is C~2 (over K) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if μ has absolutely continuous conditional measures on the unstable manifolds. The converse is also true under an additional condition on K when the system has no zero Lyapunov exponent.