Gaussian and their Subordinated Self-similar Random Generalized Fields

Abstract

A large class of generalized random fields is defined, containing random elements $F$ of $\backslash mathscr{J}'$, where $\backslash mathscr{J}'$ is the dual of the Schwartz space $\backslash mathscr{J} = \backslash mathscr{J}(\backslash mathbb{R}^\backslash nu)$. Such a generalized random field is translation-invariant if $F\backslash phi$ is the same as $F\backslash psi$ for any translate $\backslash psi$ of $\backslash phi$; it is invariant under the renormalization group with index $_\backslash kappa$ (or self-similar with index $_\backslash kappa$) if $F\backslash phi_\backslash lambda = \backslash lambda^{-\backslash alpha}F\backslash phi$ for all $\backslash lambda > 0$ and $\backslash phi \backslash in \backslash mathscr{L}$, where $\backslash phi_\backslash lambda$ is the rescaled test function $\backslash phi_\backslash lambda⊗ = \backslash lambda^{-\backslash nu}\backslash phi(x/\backslash lambda)$. Recent work of several authors has shown that self-similar generalized random fields on $\backslash mathbb{R}^\backslash nu$, and self-similar random fields on $\backslash mathbb{Z}^\backslash nu$ 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 $\backslash mathbb{R}^\backslash nu$ 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.