Journal of Mathematical Sciences

, Volume 167, Issue 6, pp 727–740 | Cite as

Random process in a homogeneous Gaussian field



We consider a random process in a spatial-temporal homogeneous Gaussian field V (q , t) with the mean E V = 0 and the correlation function W(|q q′|, |t − t′|) ≡ E[V (q, t)V (q′, t′)], where \( \bold{q} \in {\mathbb{R}^d} \), \( t \in {\mathbb{R}^{+} } \), and d is the dimension of the Euclidean space \( {\mathbb{R}^d} \). For a “density” G(r, t) of the familiar model of a physical system averaged over all realizations of the random field V, we establish an integral equation that has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for G(r, t) as r → ∞ and t → ∞.


Correlation Function Asymptotic Behavior Renormalization Group Asymptotic Expression Dyson Equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Abramowitz and I. Stegan, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Wiley, New York (1972).MATHGoogle Scholar
  2. 2.
    V. I. Alkhimov, Teor. Mat. Fiz., 139, 512–528 (2004).MathSciNetGoogle Scholar
  3. 3.
    N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantum Fields, Wiley, New York (1959).Google Scholar
  4. 4.
    R. P. Feynman, Statistical Mechanics, Benjamin, New York (1972).Google Scholar
  5. 5.
    M. Kac, Probability and Related Topics in Physical Sciences, Interscience, New York (1959).MATHGoogle Scholar
  6. 6.
    L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).MATHGoogle Scholar
  7. 7.
    L. S. Schulman, Techniques and Applications of Path Integration, Wiley, New York (1981).MATHGoogle Scholar
  8. 8.
    E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, Oxford (1939).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Moscow City University of Psychology and EducationMoscowRussia

Personalised recommendations