| This dissertation is mainly focused on the geometric properties of two kinds of anisotropic Gaussian random fields: fractional Brownian sheets and the random string processes.; Fractional Brownian sheets arise naturally in many areas, including in stochastic partial differential equations and in studies of the most visited sites of symmetric Markov processes. We prove that fractional Brownian sheets have the property of sectorial local non-determinism. By introducing a notion of dimension, called Hausdorf dimension contour, we determine the Hausdorff dimensions of the images of fractional Brownian sheets for arbitrary Borel sets. Then we provide sufficient conditions for the images to be Salem sets or to have interior points.; The random string processes are specified by a stochastic partial differential equation with different initial conditions [Funaki (1983), Mueller and Tribe (2002)]. We determine the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string processes. We also consider the Hausdorff and packing dimensions of the ranges and graphs of the strings.; We conclude this dissertation by describing some of our ongoing projects. |
