This article deals with the distribution of the view of a random environment as seen by an observer whose location at each moment is determined by the environment. The main application is in statistical fluid mechanics, where the environment consists of a random velocity field and the observer is a particle moving in the velocity field, possibly subject to molecular diffusion. Several results on such Lagrangian observations of the environment have appeared in the literature, beginning with the 1957 dissertation of J. L. Lumley. This article unites these results into a simple unified framework and rounds out the theory with new results in several directions. When the environment is homogeneous, the problem can be re-cast in terms of certain random mappings on the physical space that are based on the random location of the observer. If these mappings preserve the invariant measure on the physical space, then the view from the random location has the same distribution as the view from the origin. If these mappings satisfy the flow property and the environment is stationary, then the succession of Lagrangian observations over time forms a strictly stationary process. In particular, for motion in a homogeneous, stationary, and nondivergent velocity field, the Lagrangian velocity (the velocity of the particle) is strictly stationary, which was first observed by Lumley. In the compressible case, the distribution of a Lagrangian observation has a density with respect to the distribution of the view from the origin, and in some cases convergence in distribution of the Lagrangian observations as time tends to infinity can be shown.
Optimal estimation of Eulerian velocity field given Lagrangian observations