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

Tomasz Komorowski, and Grzegorz Krupa. "A Note on an Application of the Lasota-York Fixed Point Theorem in the Turbulent Transport Problem." Bulletin of the Polish Academy of Sciences. Mathematics 52.1 (2004): 101-113. <http://eudml.org/doc/280700>.

@article{TomaszKomorowski2004,
abstract = {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 filtrations of σ-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]).},
author = {Tomasz Komorowski, Grzegorz Krupa},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
language = {eng},
number = {1},
pages = {101-113},
title = {A Note on an Application of the Lasota-York Fixed Point Theorem in the Turbulent Transport Problem},
url = {http://eudml.org/doc/280700},
volume = {52},
year = {2004},
}

TY - JOUR
AU - Tomasz Komorowski
AU - Grzegorz Krupa
TI - A Note on an Application of the Lasota-York Fixed Point Theorem in the Turbulent Transport Problem
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2004
VL - 52
IS - 1
SP - 101
EP - 113
AB - 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 filtrations of σ-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]).
LA - eng
UR - http://eudml.org/doc/280700
ER -