On Stationarity of Lagrangian Observations of Passive Tracer Velocity in a Compressible

Here u= (u1, . . . , ud) :R d ×Ω→ Rd is the Eulerian velocity field of the flow. It is assumed to be a stationary, d-dimensional random vector field given over a certain probability space (Ω,V,P), and w(·) is a standard d-dimensional Brownian motion defined over another probability space (Σ,A,Q). Parameter κ, called the molecular diffusivity of the medium, is assumed to be nonnegative. The resulting process x(·) is considered over the product probability space (Ω×Σ,V ⊗A,P⊗Q). A question that generates considerable interest in statistical hydrodynamics is to provide the description of the long-time, large-scale asymptotics of x(·). Possible types of the trajectory behavior that may occur in the limit include Newtonian motions, diffusions, fractional diffusions and possibly Lévy flights; see [2, 6, 7, 18].