A note on diffusions in compressible environments

• Autorzy: Krupa, G.
• Języki publikacji: EN

Abstrakty

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].

Słowa kluczowe

EN

random field; diffusions in random media; Lagrangian process

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2004
• Tom: Vol. 24, Fasc. 1
• Strony: 191--210
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Krupa, G.

• Faculty of Mathematics and Natural Sciences, The Catholic University of Lublin, Lublin, Poland,

Bibliografia

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• [4] T. Komorowski and G. Krupa, On the existence of invariant measure for Lagrangian velocity in compressible environments, J. Statist. Phys. 106 (2002), pp. 635-651; Erratum, J. Statist. Phys. 109 (2002), p. 341.
• [5] T. Komorowski and G. Krupa, A note on an application of the Lasota-York fixed point theorem in the turbulent transport problem, Bull. Polish Acad. Sci. 52 (1) (2003), pp. 101-113.
• [6] T. Komorowski and G. Krupa, On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment, Ann. Appl. Probab. (2004), http://kni.kul.lublin.pl/-gkrupa/Articles/CTIM703B.ps.
• [7] A. Lasota and M. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge Univ. Press, 1985.
• [8] Yu. A. Rozanov, Stationary Random Processes, Holden-Day, 1969.
• [9] A. V. Skorokhod, σ-algebras of events on probability spaces. Similarity and factorization, Theory Probab. Appl. 36 (1990), pp. 63-73.
• [10] D. Strook and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin-Heidelberg-New York 1979.
• [11] A. S. Sznitman and M. Zerner, A law of large numbers for random walks in random environment, Ann. Probab. 27 (1999), pp. 1851-1869.