We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.
"On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment." Ann. Appl. Probab. 14 (4) 1666 - 1697, November 2004. https://doi.org/10.1214/105051604000000945