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A note on diffusions in compressible environments

Krupa G.

Probability and Mathematical Statistics

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2004

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Vol. 24, Fasc. 1

191--210

EN

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 some what similar result, but for time-dependent flows with different statistical properties, has been shown in [5].

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A note on an application of the Lasota-York fixed point theorem in the turbulent transport problem

Komorowski T., Krupa G.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 1

101-113

EN

We study a model of motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In [8] we have shown that there exists a probability measure equivalent to the underlying physical probability under which the quasi-Lagrangian velocity process, i.e. the velocity of the flow observed from the vintage point of the moving particle, is stationary and ergodic. As a consequence, we proved the existence of the mean of the quasi-Lagrangian velocity, the so-called Stokes drift of the flow. The main step in the proof was an application of the Lasota-York theorem on the existence of an invariant density for Markov operators that satisfy a lower bound condition. However, we also needed some technical condition on the statistics of the velocity field that allowed us to use the factoring property of nitrations of [sigma]-algebras proven by Skorokhod. The main purpose of the present note is to remove that assumption (see Theorem 2.1). In addition, we prove the existence of an invariant density for the semigroup of transition probabilities associated with the abstract environment process corresponding to the passive tracer dynamics (Theorem 2.7). In Remark 2.8 we compare the situation considered here with the case of steady (time independent) flow where the invariant measure need not be absolutely continuous (see [9]).

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The existence of the effective diffusivity tensor for diffusions with incompressible mixing drifts

Komorowski T., Widelski P.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

337--355

EN

In the present article we consider a model of motion of a passive tracer particle under a random, non-steady (time dependent), incompressible velocity flow in a medium with positive molecular diffusivity. We show the existence of the effective diffusivity tensor for the flow provided that its relaxation time is sufficiently small. In contrast to the previous papers [23], [6], [20] we do not assume the existence of the stationary and integrable stream matrix for the flow.