Gaussian processes, bridges and membranes extracted from selfsimilar random fields

Abstract

We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on $\mathbb{R}^d$ and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on $\mathbb{R}^d$, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on $\mathbb{R}^d$. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.