Occupation times of continuous Markov processes
Abstract (Summary)ii Craig Zirbel, Advisor We study the long-time asymptotic growth rate of mean occupation times of certain multidimensional continuous strong Markov processes. This problem for one-dimensional diffusions has been studied by many authors, also in the context of the limiting distributions of occupation times. The existing results for linear diffusions use analytical methods. They rely on Krein’s correspondence from one-dimensional theory of strings. We extend the one-dimensional results of Kasahara, Kotani and Watanabe, (1975-1982) improved by Zirbel (1997), to multidimensional diffusions which are time changed Wiener processes. The difficulties in generalizing the existing methods to higher dimensions arise from the fact that the Wiener process in higher dimensions does not revisit single points, therefore we need a new approach. The method we propose is probabilistic and it works in multiple dimensions. Our approach uses reversibility and the scaling property of the Wiener process. We find a decomposition for the mean occupation time of a reversible multidimensional diffusion in terms of a function h which reflects recurrence properties of the process and a factor depending on the function used to measure the occupation time. For time changed Wiener process having radial speed density given by a power function, we find the recurrence function h exactly. This gives us the asymptotics of mean occupation times of radially time changed d-dimensional Wiener process with speed density m(x) = c|x|?. We prove a comparison result which allows us to determine the asymptotic behavior of the function h in some non-radial cases. The bounds on hitting times combined with the comparison result allow us to determine the mean occupation times also in some non-radial cases.
School Location:USA - Ohio
Source Type:Master's Thesis
Date of Publication: