Invariant measures for passive tracer dynamics in Ornstein-Uhlenbeck flows

Let V(t,x), be a time-space stationary d-dimensional Markovian and Gaussian random field given over a probability space . Consider a diffusion with a random drift given by the stochastic differential equation , x(0)=0, where w(·) is a standard d-dimensional Brownian motion defined over another probability space . The so-called Lagrangian process, i.e. the process describing the velocity at the position of the moving particle, [eta](t):=V(t,x(t)), t[greater-or-equal, slanted]0 is considered over the product probability space . It is well known, see e.g. (Lumley, Méchanique de la Turbulence. Coll. Int du CNRS á Marseille. Ed. du CNRS, Paris; Port and Stone, J. Appl. Probab. 13 (1976) 499), that [eta](·) is stationary when the realizations of the drift are incompressible. We consider the case of fields with compressible realizations and show that there exists a probability measure, absolutely continuous with respect to , under which the Lagrangian process is stationary, provided that the velocity field V decorrelates sufficiently fast in time. Our result includes also the case [kappa]=0, i.e. motions in a random field. We prove that in the case of positive molecular diffusivity [kappa] the absolutely continuous invariant measure is unique and in fact is equivalent to . We formulate sufficient conditions on the spectrum of V that allow to claim ergodicity of the invariant measure in the case of random motions ([kappa]=0).