View source for Random field, homogeneous

A [[Random field|random field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773201.png" /> defined on a [[Homogeneous space|homogeneous space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773202.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773203.png" /> equipped with a transitive transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773204.png" /> of mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773205.png" /> into itself, and having the property that the values of the statistical characteristics of this field do not change when elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773206.png" /> are applied to their arguments. One distinguishes two different classes of homogeneous random fields: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773207.png" /> is called a homogeneous random field in the strict sense if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r0773209.png" />, the finite-dimensional probability distribution of its values at any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732010.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732011.png" /> coincides with that of its values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732018.png" /> is called a homogeneous random field in the wide sense.

An important special case is that of a homogeneous random field on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732019.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732020.png" /> (or on the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732021.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732022.png" /> with integral coordinates), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732023.png" /> is the group of all parallel translations. Sometimes the term  "homogeneous random field"  is reserved for a field of this type. A homogeneous random field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732024.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732025.png" /> the group of all isometric transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732026.png" /> (generated by parallel translations, rotations and reflections) is often called an isotropic homogeneous random field.

The concept of a homogeneous random field is a natural generalization of that of a [[Stationary stochastic process|stationary stochastic process]]: in both cases, the field and the covariance function admit a spectral decomposition (cf. [[Spectral decomposition of a random function|Spectral decomposition of a random function]]) of special kind (see, for example, [[#References|[1]]]–[[#References|[5]]]). Homogeneous random fields and some of their generalizations often arise in questions of an applied nature. In particular, in the statistical theory of turbulence, an important role is played by isotropic homogeneous (scalar and vector) random fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077320/r07732027.png" />, as well as by so-called simultaneously locally homogeneous and locally isotropic random fields (that is, fields with homogeneous and isotropic increments), which are simple generalizations of isotropic homogeneous fields (see [[#References|[4]]], for example). Moreover, in the modern theory of physical quantum fields and in statistical physics there are applications of the theory of generalized homogeneous random fields, which include homogeneous random fields as a special case (see [[Random field, generalized|Random field, generalized]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Yaglom,   "Second-order homogeneous random fields" , ''Proc. 4-th Berkeley Symp. Math. Stat. Probab.'' , '''2''' , Univ. California Press  (1961)  pp. 593–622</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,   "Group representations and applied probability" , Methuen  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Yadrenko,   "Spectral theory of random fields" , Optim. Software  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.S. Monin,   A.M. Yaglom,   "Statistical fluid mechanics" , '''2''' , M.I.T.  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.M. Yaglom,   "Correlation theory of stationary and related random functions" , '''1–2''' , Springer  (1987)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Ramm,   "Random fields: estimation theory" , Longman &amp; Wiley  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.-O. Georgii,   "Gibbs measures and phase transitions" , de Gruyter  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Albeverio,   R. Høegh-Krohn,   "Homogeneous random fields and statistical mechanics"  ''J. Funct. Anal.'' , '''19'''  (1975)  pp. 242–272</TD></TR></table>