This research will study two different classes of problems for Markov processes. The first is a study of the asymptotic distribution of certain generalized occupation time formulas for special classes of Markov processes. The investigator has recently obtained generalized arcsine laws for occupation times of one dimensional Levy processes in joint work with M.J.Sharpe. He will attempt to find higher dimensional analogues of these results. The second class of problems deals with the potential theory of a general Markov process. In a recent monograph the principal investigator has shown that for many purposes excessive measures are more fundamental than excessive functions. He plans to continue his investigation of the potential theory of excessive measures. In particular the connection with capicitary measures and excursion theory. This research is in the area of Markov processes. A Markov process is a mathematical model of how certain random phenomena evolve in time. Examples are the diffusion of a gas or liquid, or the motion of a collidal particle in a liquid subject to molecular bombardment. This last example is called Brownian motion. Surprisingly the mathematical model for Brownian motion also turns out to be of fundamental importance in potential theory which encompasses electrostatic potentials and equilibrium temperature distributions in solids among other things.
