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

Tomasz Komorowski ¹   ; Grzegorz Krupa ²

¹ Institute 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
² Faculty of Mathematics and Natural Sciences The Catholic University of Lublin Al. Racławickie 14 20-950 Lublin, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 1, pp. 101-113

Résumé

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}).

Détail

DOI : 10.4064/ba52-1-11

Keywords: study model motion passive tracer particle turbulent flow strongly mixing time variable kokr jsp have shown there exists probability measure equivalent underlying physical probability under which quasi lagrangian velocity process velocity flow observed vintage point moving particle stationary ergodic consequence proved existence mean quasi lagrangian velocity so called stokes drift flow main step proof application lasota york theorem existence invariant density markov operators satisfy lower bound condition however needed technical condition statistics velocity field allowed factoring property filtrations sigma algebras proven skorokhod main purpose present note remove assumption see theorem addition prove existence invariant density semigroup transition probabilities associated abstract environment process corresponding passive tracer dynamics theorem remark compare situation considered here steady time independent flow where invariant measure absolutely continuous see kokr aap