## The Annals of Probability

### Gaussian and their Subordinated Self-similar Random Generalized Fields

R. L. Dobrushin

#### Abstract

A large class of generalized random fields is defined, containing random elements $F$ of $\mathscr{J}'$, where $\mathscr{J}'$ is the dual of the Schwartz space $\mathscr{J} = \mathscr{J}(\mathbb{R}^\nu)$. Such a generalized random field is translation-invariant if $F\phi$ is the same as $F\psi$ for any translate $\psi$ of $\phi$; it is invariant under the renormalization group with index $_\kappa$ (or self-similar with index $_\kappa$) if $F\phi_\lambda = \lambda^{-\alpha}F\phi$ for all $\lambda > 0$ and $\phi \in \mathscr{L}$, where $\phi_\lambda$ is the rescaled test function $\phi_\lambda(x) = \lambda^{-\nu}\phi(x/\lambda)$. Recent work of several authors has shown that self-similar generalized random fields on $\mathbb{R}^\nu$, and self-similar random fields on $\mathbb{Z}^\nu$ which can be constructed from them, arise naturally in problems of statistical mechanics and limit laws of probability theory. They generalize the theory of stable distributions. Here the class of all translation-invariant self-similar Gaussian generalized random fields on $\mathbb{R}^\nu$ is completely described. By the discretization of such fields the class of self-similar Gaussian fields with discrete arguments (found by Sinai) is extended. Finally, a class of generalized random fields subordinated to the self-similar translation-invariant Gaussian ones is constructed. These non-Gaussian generalized random fields are Wick powers (multiple Ito integrals) of the Gaussian ones.

#### Article information

Source
Ann. Probab., Volume 7, Number 1 (1979), 1-28.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995145

Digital Object Identifier
doi:10.1214/aop/1176995145

Mathematical Reviews number (MathSciNet)
MR515810

Zentralblatt MATH identifier
0392.60039

JSTOR