Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

Abstract

We study the spectral properties of spatial and spatiotemporal Gaussian random fields defined as the solutions to stochastic elliptic, parabolic, and hyperbolic fractional pseudodifferential equations on compact fractal domains. The fractal dimension of the domain modifies the asymptotic properties of the eigenvalues that define the pure point spectra of the covariance functions of the solutions and their Karhunen-Loève-type expansions. The eigenfunction systems involved constitute orthogonal bases of the corresponding trace spaces on fractal sets. The Hölder exponent of the sample paths of the random fields is computed in terms of the fractional order of mean-quadratic variation on their increments. Such an exponent also depends on the Hausdorff dimension of the domain.

Keywords:

• Fractal domains
• Fractional generalized random fields
• Karhunen-Loève representations
• Stochastic fractional pseudodifferential equations

Mathematics Subject Classification:

• Primary 60G60
• Secondary 60G20

ACKNOWLEDGMENT

This work is partially supported by project BFM2002-01836 of the DGI, Spain, and by the Australian Research Council grants LX0348297 and DP0559807.