The Annals of Probability

Gaussian and their Subordinated Self-similar Random Generalized Fields

R. L. Dobrushin

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G20: Generalized stochastic processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Generalized random fields renormalization groups self-similar Gaussian fields


Dobrushin, R. L. Gaussian and their Subordinated Self-similar Random Generalized Fields. Ann. Probab. 7 (1979), no. 1, 1--28. doi:10.1214/aop/1176995145.

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