Electronic Communications in Probability Surface Stretching for Ornstein Uhlenbeck Velocity Fields

The present note deals with large time properties of the Lagrangian trajectories of a turbulent ow in IR 2 and IR 3. We assume that the ow is driven by an incompressible time-dependent random velocity eld with Gaussian statistics. We also assume that the eld is homogeneous in space and stationary and Markovian in time. Such velocity elds can be viewed as (possibly innnite dimensional) Ornstein-Uhlenbeck processes. In d spatial dimensions we established the (strict) positivity of the sum of the largest d?1 Lyapunov exponents. As a consequences of this result, we prove the exponential stretching of surface areas (when d = 3) and of curve lengths (when d = 2) which connrms conjectures found in the theory of turbulent ows.