| In this thesis, a class of processes called tempered stable subordinators is defined. By subordinating Brownian motion {lcub}Bt : t ≥ 0{rcub} with independent tempered stable subordinators {lcub}Zt : t ≥ 0{rcub}, we can get a class of Levy processes Xt := BZtw (o) in Rd (d ≥ 2). Some well known processes such as symmetric alpha-stable process and relativistic alpha-stable processes are special examples in this class. Recently for a bounded open set D with C1,1 boundary, the Green functions of the killed symmetric alpha-stable processes and relativistic alpha-stable processes upon leaving D were studied and sharp estimates were established. Under some assumptions on the Levy measure of the subordinator {lcub}Zt : t ≥ 0{rcub}, we can get sharp estimates on the Green function of the process {lcub} XDt {rcub}. And if the Levy measure of the subordinator satisfies some extra conditions, we can establish the Harnack inequality and boundary Harnack principle for all processes in this class in a unified way. Then the nonnegative harmonic functions of the symmetric alpha-stable processes and relativistic alpha-stable process have these properties as special examples. In the first chapter, we give a brief introduction to Levy processes and subordination. Then the definition of the Green functions and harmonic functions are recalled and discussed in the second chapter. In the next two chapters, sharp estimates of the Green functions of {lcub} XDt {rcub} and the boundary Harnack principle are proved. |
