Although a lot is known about symmetric stable processes and Levy processes, until recently, little was known about the counterparts to some of the deep results and fine properties of Brownian motion and diffusion processes. The PI, jointly with Zhen-Qing Chen, has recently obtained sharp estimates on the Green function and Poisson kernel of symmetric stable processes on bounded smooth domains. And these estimates have been applied to study the intrinsic ultracontractivity and conditional gauge theorem for symmetric stable processes on smooth domains. The PI intends to continue this study of the fine properties of symmetric stable processes first and then to investigate the fine properties of more general Levy processes or Markov processes. More precisely, he intends to do the following: 1) study the Martin boundary and integral representations of positive superharmonic functions of symmetric stable processes; 2) establish the intrinsic ultracontractivity and conditional gauge theorem for symmetric stable processes on rough domains; 3) establish the boundary Harnack principle for symmetric stable processes on domains rougher than Lipschitz domains; 4) find some nontrivial estimates on the first eigenvalue of the generator of the killed symmetric stable processes; and 5) generalize the results proposed above to more general Levy processes.
One of the most important classes of Markov processes is the Levy processes or processes with independent stationary increments. And one of the most important classes of Levy processes is the symmetric stable processes. Brownian motion is the only symmetric stable process with continuous sample paths. Brownian motion plays a central role in modern probability theory and has been intensively studied. In the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes and other Levy processes. Stable processes and Levy processes are now widely used in physics, operations research, queueing theory, mathematical finance and risk estimation. These widespread applications are the motivation for this investigator to study the fine properties of Levy processes. The principal investigator expects that this proposed study will make symmetric stable processes and Levy processes applicable to a wider range of practical areas and to more difficult problems.
