Invariant measures for passive tracer dynamics in Ornstein–Uhlenbeck flows

Abstract

Let V (t, x ) , (t, x )∈ R× R d be a time–space stationary d-dimensional Markovian and Gaussian random field given over a probability space T 0≔(Ω, V, P) . Consider a diffusion with a random drift given by the stochastic differential equation d x (t)= V (t, x (t)) dt+ 2κ d w (t) , x (0)= 0 , where w (·) is a standard d-dimensional Brownian motion defined over another probability space T 1≔(Σ, W, W) . The so-called Lagrangian process, i.e. the process describing the velocity at the position of the moving particle, η(t)≔ V (t, x (t)) , t⩾0 is considered over the product probability space T 0⊗ T 1 . 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 η(·) 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 P⊗ W , under which the Lagrangian process is stationary, provided that the velocity field V decorrelates sufficiently fast in time. Our result includes also the case κ=0, i.e. motions in a random field. We prove that in the case of positive molecular diffusivity κ the absolutely continuous invariant measure is unique and in fact is equivalent to P⊗ W . We formulate sufficient conditions on the spectrum of V that allow to claim ergodicity of the invariant measure in the case of random motions ( κ=0).