The First Eigenvalue Of Transient Birth-Death Processes Of L~p Space

Birth-death process, as a typical of continuous-time Markov process, plays an important role in the theory of stochastic processes, and it has a wide range of applications in the natural sciences, biology, physics, queuing theory and other fields. Motivated by the theory of ergodic theory of Markov processes by Professor Mu-Fa Chen and other probabilistic scholars, the eigenvalue problem for birth-death process has been paid much attention in recent years. The reader can refer to the book [12] and the references therein. Let X={Xt, t≥ 0} be a homogeneous continuous-time Markov process defined on the probability space (Ω. F, P) and with value in Z+ ={0,1,…}, and let Q={pij), i,j € Z+, be its standard Q matrix. If 9i,i+1= bi (i≥ 0), qi,i-1= αi (i≥ 1), and qi,j - 0 for all |i-j|≥ 2, then X is called a birth-death process. In this paper, we briefly call birth-death process with birth rates bi> 0 (i≥ 0). and death rates αi> 0 (i≥ 1). It is easy to see that birth-death process is a symmetric Markov process, and its symmetric measure μ= (μt)i≥o is given by μ0= 1, Furthermore, the birth-death process is transient if and only if it is ergodic if and only if For other criterion about the birth-death process we can refer to [10].The purpose of this thesis is to study the first eigenvalue and related topics for transient birth-death processes on IP space. The thesis consists of the following four chapters.In Chapter One, we first introduce some background for birth-death process, and then recall related developments. We also briefly present our main results and methods.Chapter Second, mainly based on [14], is devoted to the first eigenvalue of birth-death process on Lp space. In particular, the upper and lower bounds of variational formulas, one basic estimate and approximating procedure about the first eigenvalue λ0p of birth-death processes on Lp space are established. We will prove the monotonic property of the first eigenvalue λ0p with respect to p, and present some examples to illustrate the power of our basic estimates.In Chapter Three, we focus on Poincare-type inequality for transient birth-death processes. More explicit, explicit bounds of optimal constants AB in Poincare-type inequalities for transient birth-death processes are presented. As applications, we will also consider Nash inequality and the logarithmic Sobolev inequality. Al-though the approach here is similar to [11], we point out some essential difference between transient birth-death process and ergodic one. In particular, we conclude that for transient birth-death process, the Poincare inequality implies Nash inequal-ity and logarithmic Soboley inequality. Such assertion is completely different from that in the ergodic settings (see [12, Table 1.4] and [37]). Furthermore, we also show that the dual approach is not efficient for the study of Poincare-type inequalities for transient birth-death processes. The equivalence between Poincare-type inequalities for transient birth-death processes with Dirichlet boundary both at 0 and ∞ and those with Dirichlet boundary only at 0 is given.Chapter Four, the last chapter of our thesis is a summary of all the result-s.Here,we also present some interesting questions for further research.