The Annals of Applied Probability

On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment

Tomasz Komorowski and Grzegorz Krupa

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We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.

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Ann. Appl. Probab., Volume 14, Number 4 (2004), 1666-1697.

First available in Project Euclid: 5 November 2004

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 60G44: Martingales with continuous parameter

Random field diffusions in random media Lagrangian process invariant measure


Komorowski, Tomasz; Krupa, Grzegorz. On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment. Ann. Appl. Probab. 14 (2004), no. 4, 1666--1697. doi:10.1214/105051604000000945.

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  • Adler, R. J. (1990). An Introduction to Continuity Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA.
  • Avellaneda, M. and Majda, A. J. (1990). Mathematical models with exact renormalization for turbulent transport. Comm. Math. Phys. 131 381–429.
  • Bolthausen, E. and Sznitman, A. S. (2002). On the static and dynamic points of views for certain random walks in random environment. Methods Appl. Anal. 9 345–376.
  • Cranston, M. and Zhao, Z. (1987). Conditional transformation of drift formula and potential theory for $\frac12\Delta+b(\cdot)\cdot\nabla$. Comm. Math. Phys. 112 613–625.
  • Davis, R. E. (1982). On relating Eulerian and Lagrangian velocity statistics: Single particles in homogeneous flows. J. Fluid Mech. 114 1–26.
  • Fannjiang, A. (1998). Anomalous diffusion in random flows. In Mathematics of Multiscale Materials (K. M. Golden, G. R. Grimmett, R. D. James, G. W. Milton and P. N. Sen, eds.) 81–89. Springer, Berlin.
  • Fannjiang, A. and Komorowski, T. (2000). The fractional Brownian motion limit for motions in turbulence. Ann. Appl. Probab. 10 1100–1120.
  • Komorowski, T. (2001). Stationarity of Lagrangian velocity in compressible environments. Comm. Math. Phys. 228 417–434.
  • Komorowski, T. and Krupa, G. (2002). On the existence of invariant measure for Lagrangian velocity in compressible environments. J. Statist. Phys. 106 635–651. [Erratum 109 341.]
  • Komorowski, T. and Olla, S. (2003). Invariant measures for passive tracer dynamics in Ornstein–Uhlenbeck flows. Stochastic Process. Appl. 105 139–173.
  • Lasota, A. and Mackey, M. (1985). Probabilistic Properties of Deterministic Systems. Cambridge Univ. Press.
  • Lumley, J. L. (1962). The mathematical nature of the problem of relating Lagrangian and Eulerian statistical functions in furbulence. Méchanique de la Turbulence. Coll. Int du CNRS á Marseille. Ed. du CNRS, Paris.
  • Olla, S. (1994). Homogenization of diffusion processes in random fields. Manuscript of Centre de Mathématiques Appliquées.
  • Port, S. C. and Stone, C. (1976). Random measures and their application to motion in an incompressible fluid. J. Appl. Probab. 13 499–506.
  • Skorokhod, A. V. (1990). $\sigma$-algebras of events on probability spaces. Similarity and factorization. Theory Probab. Appl. 36 63–73.
  • Strook, D. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
  • Sznitman, A. S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869.
  • Zaslavsky, G. M. (1994). Fractional kinetic equation for Hamiltonian chaos. Phys. D 76 110–122.
  • Zirbel, C. (2001). Lagrangian observations of homogeneous random environments. Adv. in Appl. Probab. 33 810–835.