Drift estimation for Brownian flows

The problem of estimating the drift of a stochastic flow given Lagrangian observations is an estimation problem for a multidimensional diffusion with a degenerate diffusion matrix. The maximum-likelihood estimator of the constant drift is considered. A long-time asymptotic of its mean-square error (MSE) is computed. It is shown that the time-space average of the observed Lagrangian velocities has the same asymptotic. These estimators are compared to the least-squares estimator based on Eulerian data. In the most important, for applications, two-dimensional case the Lagrangian estimator is typically preferable for incompressible flows, while for flows close to potential the Eulerian estimator is better.