Gaussian random fields related to Lévy's Brownian motion : representations and expansions

Graduation date: 2013This dissertation examines properties and representations of several isotropic Gaussian random fields in the unit ball in d-dimensional Euclidean space. First we consider Lévy's Brownian motion. We use an integral representation for the covariance function to find a new expansion for Lévy's Brownian motion as an infinite linear combination of independent standard Gaussian random variables and orthogonal polynomials. \ud \ud Next we introduce a new family of isotropic Gaussian random fields, called the p-processes, of which Lévy's Brownian motion is a special case. Except for Lévy's Brownian motion the p-processes are not locally stationary. All p-processes also have a representation as an infinite linear combination of independent standard Gaussian random variables. \ud \ud We use these expansions of the random fields to simulate Lévy's Brownian motion and the p-processes along a ray from the origin using the Cholesky factorization of the covariance matrix