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

Consider a random cocycle Phi on a separable in finite-dimensional Hilbert space preserving a probability measure mu, which is supported on a random compact set K. We show that if Phi is C-2 (over K) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if mu 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.Mathematics, AppliedMathematicsSCI(E)0ARTICLE94123-41553