Diffusions with Gaussian Drifts

  • Tomasz Komorowski
  • Claudio Landim
  • Stefano Olla
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)


We continue our investigation of the passive tracer model introduced in Chap.  11 under an additional assumption that the drift is Gaussian. In this case the generator of the environment process satisfies the graded sector condition introduced in Sect.  2.7.4. Using this approach we reprove first the central limit theorem for diffusions whose drift has a finite Péclet number. This result can be extended to some classes of time dependent flows whose Péclet number are infinite by taking into account time decorrelation properties of the drift. We apply the variational principles derived in Chap.  4 to prove that the sufficient conditions for the central limit theorem obtained in this way are in some sense optimal. We give examples of families of isotropic flows for which the motion of a tracer is superdiffusive.


Variational Principle Central Limit Theorem Stochastic Differential Equation Spectral Measure Environment Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Adler RJ (1990) An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS lect notes monogr ser, vol 12. Inst Math Stat, Hayward zbMATHGoogle Scholar
  2. Avellaneda M, Majda AJ (1990) Mathematical models with exact renormalization for turbulent transport. Commun Math Phys 131(2):381–429 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Avellaneda M, Majda A (1992a) Mathematical models with exact renormalization for turbulent transport. II. Fractal interfaces, non-Gaussian statistics and the sweeping effect. Commun Math Phys 146(1):139–204 MathSciNetzbMATHCrossRefGoogle Scholar
  4. Avellaneda M, Majda AJ (1992b) Superdiffusion in nearly stratified flows. J Stat Phys 69(3–4):689–729 MathSciNetzbMATHCrossRefGoogle Scholar
  5. Batchelor GK (1982) The theory of homogeneous turbulence. Cambridge University Press, Cambridge. Reprint, Cambridge Science Classics. zbMATHGoogle Scholar
  6. Ben Arous G, Owhadi H (2002) Super-diffusivity in a shear flow model from perpetual homogenization. Commun Math Phys 227(2):281–302 MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bernardin C (2004) Fluctuations in the occupation time of a site in the asymmetric simple exclusion process. Ann Probab 32(1B):855–879 MathSciNetzbMATHCrossRefGoogle Scholar
  8. Carmona RA, Xu L (1997) Homogenization for time-dependent two-dimensional incompressible Gaussian flows. Ann Appl Probab 7(1):265–279 MathSciNetzbMATHCrossRefGoogle Scholar
  9. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York zbMATHCrossRefGoogle Scholar
  10. Fannjiang AC (2000) Phase diagram for turbulent transport: sampling drift, eddy diffusivity and variational principles. Physica D 136(1–2):145–174 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Fannjiang A, Komorowski T (1999b) Turbulent diffusion in Markovian flows. Ann Appl Probab 9(3):591–610 MathSciNetzbMATHCrossRefGoogle Scholar
  12. Fannjiang A, Komorowski T (2000a) Fractional Brownian motions and enhanced diffusion in a unidirectional wave-like turbulence. J Stat Phys 100(5–6):1071–1095 MathSciNetzbMATHCrossRefGoogle Scholar
  13. Fannjiang A, Komorowski T (2000b) Fractional Brownian motions in a limit of turbulent transport. Ann Appl Probab 10(4):1100–1120 MathSciNetzbMATHGoogle Scholar
  14. Fannjiang A, Komorowski T (2001/2002) Diffusive and nondiffusive limits of transport in nonmixing flows. SIAM J Appl Math 62(3):909–923 (electronic) MathSciNetCrossRefGoogle Scholar
  15. Fannjiang A, Komorowski T (2002a) Diffusion in long-range correlated Ornstein-Uhlenbeck flows. Electron J Probab 7(20):22 (electronic) MathSciNetGoogle Scholar
  16. Friedman A (1975) Stochastic differential equations and applications, vol 1. Probability and mathematical statistics, vol 28. Academic Press [Harcourt Brace Jovanovich Publishers], New York zbMATHGoogle Scholar
  17. Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 224, 2nd edn. Springer, Berlin zbMATHCrossRefGoogle Scholar
  18. Janson S (1997) Gaussian Hilbert spaces. Cambridge tracts in mathematics, vol 129. Cambridge University Press, Cambridge zbMATHCrossRefGoogle Scholar
  19. Komorowski T, Nieznaj E (2008) On superdiffusive behavior of a passive tracer in a random flow. Physica D 237(24):3377–3381 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Komorowski T, Olla S (2002) On the superdiffusive behavior of passive tracer with a Gaussian drift. J Stat Phys 108(3–4):647–668 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Komorowski T, Olla S (2003b) On the sector condition and homogenization of diffusions with a Gaussian drift. J Funct Anal 197(1):179–211 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Komorowski T, Ryzhik L (2007a) On asymptotics of a tracer advected in a locally self-similar, correlated flow. Asymptot Anal 53(3):159–187 MathSciNetzbMATHGoogle Scholar
  23. Komorowski T, Ryzhik L (2007b) Passive tracer in a slowly decorrelating random flow with a large mean. Nonlinearity 20(5):1215–1239 MathSciNetzbMATHCrossRefGoogle Scholar
  24. Koralov L (1999) Transport by time dependent stationary flow. Commun Math Phys 199(3):649–681 MathSciNetzbMATHCrossRefGoogle Scholar
  25. Landim C, Quastel J, Salmhofer M, Yau HT (2004b) Superdiffusivity of asymmetric exclusion process in dimensions one and two. Commun Math Phys 244(3):455–481 MathSciNetzbMATHCrossRefGoogle Scholar
  26. Majda AJ, Kramer PR (1999) Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys Rep 314(4–5):237–574 MathSciNetCrossRefGoogle Scholar
  27. Matheron G, DeMarsily G (1980) Is transport in porous media always diffusive? A counterexample. Water Resou Bull 16:901–917 CrossRefGoogle Scholar
  28. Owhadi H (2004) Averaging versus chaos in turbulent transport? Commun Math Phys 247(3):553–599 MathSciNetzbMATHCrossRefGoogle Scholar
  29. Rozanov YA (1967) Stationary random processes. Holden-Day, San Francisco. Translated from the Russian by A Feinstein zbMATHGoogle Scholar
  30. Sethuraman S (2000) Central limit theorems for additive functionals of the simple exclusion process. Ann Probab 28(1):277–302 MathSciNetzbMATHCrossRefGoogle Scholar
  31. Tóth B, Valko B (2010) Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2. arXiv:1012.5698
  32. Zhang Q, Glimm J (1992) Inertial range scaling of laminar shear flow as a model of turbulent transport. Commun Math Phys 146(2):217–229 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomasz Komorowski
    • 1
    • 2
  • Claudio Landim
    • 3
    • 4
  • Stefano Olla
    • 5
  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Instituto de Matemática (IMPA)Rio de JaneiroBrazil
  4. 4.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  5. 5.CEREMADE, CNRS UMR 7534Université Paris-DauphineParisFrance

Personalised recommendations