Probability Measure and Characteristic Functional

  • Wolfgang KollmannEmail author


The variables for the complete description of the statistical properties of turbulence in incompressible Newtonian fluids and the associated equations governing them are set up and their fundamental properties, such as mathematical type, linearity/nonlinearity, solvability, etc., are discussed in the present section.


  1. 1.
    Cartier, P., DeWitt-Morette, C.: A new perspective on functional integration. J. Math. Phys. 36, 2137–2340 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    DeWitt-Morette, C., Cartier, P., Folacci, A.: Functional Integration. Plenum Press, New York, London (1997)CrossRefGoogle Scholar
  3. 3.
    Cartier, P., DeWitt-Morette, C.: Functional Integration: Action and Symmetries. Cambridge University Press, Cambridge U.K (2006)CrossRefGoogle Scholar
  4. 4.
    Simon, B.: Functional Integration and Quantum Physics. AMS Chelsea Publication, Providence, Rhode Island (2004)CrossRefGoogle Scholar
  5. 5.
    Klauder, J.R.: A Modern Approach to Functional Integration. Birkhaeuser/Springer, New York (2010)zbMATHGoogle Scholar
  6. 6.
    Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publication, Dordrecht (1988)CrossRefGoogle Scholar
  7. 7.
    Dalecky, Y.L., Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Space. Kluwer Academic Publication, Dordrecht (1991)CrossRefGoogle Scholar
  8. 8.
    Shilov, G.E., Gurevich, B.L.: Integral, Measure and Derivative: A Unified Approach. Prentice Hall Inc., Englewood Cliffs (1966)zbMATHGoogle Scholar
  9. 9.
    Bogachev, V.I.: Measure Theory, vol. 1. Springer, New York (2006)Google Scholar
  10. 10.
    Yamasaki, Y.: Measures on Infinite Dimensional Spaces. World Scientific, Singapure (1985)CrossRefGoogle Scholar
  11. 11.
    Conway, J.: A Course in Functional Analysis. Springer, New York (1990)Google Scholar
  12. 12.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)zbMATHGoogle Scholar
  13. 13.
    Minlos, R.A.: Cylinder sets. In: Hazewinkel, M. (ed.) Encyclopdia of Mathematics. Springer, New York (2001)Google Scholar
  14. 14.
    Skorohod, A.V.: Integration in Hilbert Space. Springer, New York (1974)CrossRefGoogle Scholar
  15. 15.
    Bogachev, V.I.: Gaussian Measures, Mathematical surveys and monographs, p. 62. American Mathematical Society (1998)Google Scholar
  16. 16.
    Sagaut, P., Cambron, C.: Homogeneous Turbulence Dynamics. Cambridge University Press, New York (2008)CrossRefGoogle Scholar
  17. 17.
    Kuksin, S., Shirikyan, A.: Mathematics of Two-dimensional Turbulence, Cambridge Tracts in Mathematics, vol. 194. Cambridge University Press, U.K. (2012)CrossRefGoogle Scholar
  18. 18.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)zbMATHGoogle Scholar
  19. 19.
    Cameron, R.H., Martin, W.T.: Transformations of Wiener integrals under a general class of linear transformations. Trans. Am. Math. Soc. 58, 184–219 (1945)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cameron, R.H., Martin, W.T.: Evaluation of various Wiener integrals by use of certain Sturm-Liouville differential equations. Bull. Am. Math. Soc. 51, 73–90 (1945)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hartmann, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  22. 22.
    Duffy, D.G.: Greenś Functions with Applications. Chapman & Hall/CRC, Boca Raton, Florida (2001)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

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