A NOTE ON DIFFUSIONS IN COMPRESSIBLE ENVIRONMENTS
Abstract: We study the equation of a motion of a passive tracer in a time-independent
turbulent flow in a medium with a positive molecular diffusivity. In [6] the authors have
shown the existence of an invariant probability measure for the Lagrangian velocity
process. This measure is absolutely continuous with respect to the underlying physical
probability for the Eulerian flow. As a result the existence of the Stokes drift has
been proved. The results of [6] were derived under some technical condition on the
statistics of the Eulerian velocity field. This condition was crucial in the proof in [6].
However, in applications it is difficult to check whether the velocity field satisfies this
condition.
In this note we show that the main result of [6] can be stated also without the
above-mentioned technical assumption. A somewhat similar result, but for time-dependent
flows with different statistical properties, has been shown in [5].
2000 AMS Mathematics Subject Classification: Primary 60F17, 35B27; Secondary
60G44.
Key words and phrases: Random field, diffusions in random media, Lagrangian process.