A Note on an Application of the Lasota–York Fixed Point Theorem in the Turbulent Transport Problem

Volume 52 / 2004

Tomasz Komorowski, Grzegorz Krupa Bulletin Polish Acad. Sci. Math. 52 (2004), 101-113 MSC: Primary 60F17, 35B27; Secondary 60G44. DOI: 10.4064/ba52-1-11

Abstract

We study a model of motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In {kokr-jsp} 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 filtrations 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 {kokr-aap}).

Authors

  • Tomasz KomorowskiInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21
    Śniadeckich 8
    00-956 Warszawa, Poland
    and
    nstitute of Mathematics
    M. Curie-Sk/lodowska University
    Pl. M. Curie Sk/lodowskiej 1
    20-031 Lublin, Poland
    e-mail
  • Grzegorz KrupaFaculty of Mathematics and Natural Sciences
    The Catholic University of Lublin
    Al. Racławickie 14
    20-950 Lublin, Poland
    e-mail

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