Dimension results for two random fractals

Random fractals are sets generated by random processes that exhibit fractal properties. These properties can be described by proper fractal dimensions. Hausdor dimension and packing dimension are two fractal dimensions that are of most interest. In this thesis two different random fractals are investigated and their fractal dimensions are computed. The first random fractal studied is the image of an additive Levy process. Let X = {X (t) : t ϵ RN +} denote an additive Levy process and F Ϲ RN + a Borel set. We define a new packing dimension profile Dim and show that the packing dimension of the image set X (F) is equal to Dim (F) almost surely. Moreover we show that our packing dimension profile gives probabilistic interpretations to the packing dimension profiles defined by Falconer and Howroyd in their study of orthogonal projection. The second random fractal studied is the limsup random fractal. A limsup random fractal is generated by limsup operations and is a model for many other random sets including fast points and thick points of a Brownian motion. We study limsup random fractals defined on the boundary of a spherically symmetric tree and estimate the probability that a limsup random fractal hits an independent fractal percolation set. By a codimension argument, we compute the Hausdor dimension of limsup random fractals and packing dimension of fractal percolation sets. This solves in part a problem introduced by Khoshnevisan, Peres, and Xiao.