The Transformation And ρ-dirichlet Form Of Birth-Death Processes

The purpose of this thesis is to discuss applications of h-transformation and space transformation for the first eigenvalue of birth-death processes, and also to present some sharp estimates for the first eigenvalue of the associated p-Dirichlet forms.In Chapter One, we give sorne background about birth-death processes. In particular, we focus on the introduction for h-transformation, space transformation and p-Dirichlet form of birth-death processes.In Chapter Two, we obtain some explicit estimates for the eigenvalue λO,∞ D,D with Dirichlet boundary at 0 and ∞ of transient birth-death processes, by adopting the proper h-transformation. Furthermore, the h-transformation approach applies to study the associated Poincare-type inequalities. As a typical application, the N ash inequality is also investigated.Chapter Three is concerned with the space transforiilation of birth-death pro-esses. We establish the relationship between the first eigenvalue λO,N N,D with Dirichlet boundary at N of transient birth-death processes and the first eigenvalue λO,N D,N* with Dirichlet boundary at 0 of ergodic birth-death processes. With the help of this rela-tionship, we verify the famous Mu-Fa Chen’s variational formulas of λO,N N,D in a new way.In Chapter Four, we obtain some sharp upper and lower bounds for the first eigenvalue on the space Lp(μ) with p>1,where and bi (i≥0) is birth rate of birth-death process,μ=(μi)i≥o is the associated symmetric measure.In Chapter Five, we summary main results of our thesis, and also present some questions for future research work.