Elliptic gaussian random processes

Abstract

We study the Gaussian random fields indexed by ℝd whose covariance is defined in all generality as the parametrix of an elliptic pseudo-differential operator with minimal regularity asumption on the symbol. We construct new wavelet bases adapted to these operators; the decomposition of the field on this corresponding basis yields its iterated logarithm law and its uniform modulus of continuity. We also characterize the local scalings of the field in term of the properties of the principal symbol of the pseudodifferential operator. Similar results are obtained for the Multi-Fractional Brownian Motion.