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On the Asymptotic Behavior of an Ornstein-Uhlenbeck Process with Random Forcing

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Abstract

We study the asymptotic behavior of an inertial tracer particle in a random force field. We show that there exists a probability measure, under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic. This measure is equivalent to the underlying probability for the Eulerian flow. As a consequence of the above we obtain the law of large numbers for the trajectory of the tracer. Moreover, we prove also some decorrelation properties of the velocity of the particle, which lead to the existence of a non-degenerate asymptotic covariance tensor.

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Authors and Affiliations

  1. Institute of Mathematics, UMCS, Lublin, Poland

    T. Komorowski

  2. Faculty of Mathematics and Natural Sciences, The Catholic University of Lublin, Lubin, Poland

    G. Krupa

Authors
  1. T. Komorowski
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  2. G. Krupa
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Correspondence to T. Komorowski.

Additional information

Communicated by J.L. Lebowitz

The research of both authors was supported by the Polish Committee for Scientific Research (KBN) grant No. 2PO3A03123.

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Komorowski, T., Krupa, G. On the Asymptotic Behavior of an Ornstein-Uhlenbeck Process with Random Forcing. Commun. Math. Phys. 261, 517–543 (2006). https://doi.org/10.1007/s00220-005-1453-z

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  • DOI: https://doi.org/10.1007/s00220-005-1453-z

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