【作者】 吴莺；

【导师】 黄志远；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2005， 博士

【摘要】 本论文主要研究了两个方面的问题。一方面重点研究了Lévy-Meixner白噪声分析。首先,我们研究了一维空间的Lévy-Meixner多项式及其母函数,然后对具有有限各阶矩的一维无穷可分分布进行“一步提升”,得到无穷维空间S^*（IR^d）上的Lévy白噪声测度;对Lévy-Meixner多项式的母函数进行“一步提升”,构造了重正化核（或Wick张量幂）.在Lévy-Meixner白噪声空间,我们给出了重正化核统一的,简单且深刻的显式表示。而且,我们定义了关于“对角线”迹的内积,从而获得了交互作用Fock空间上的Lévy-Meixner白噪声泛函的统一的显式的混沌分解。而这个交互作用性质完全由我们所定义的交互作用指数决定,且经典的Fock空间可以看作交互作用Fock空间的商空间。这种方法让我们很容易地得到Lévy-Meixner场算子的一般表示形式。接着我们构造了Lévy-Meixner白噪声分析框架,在此框架下我们研究了Lévy-Meixner场算子和点态Lévy-Meixner场。另一方面研究了q-Lévy-Meixner多项式及其母函数。首先,我们定义了q-Lévy-Meixner多项式及其母函数,然后给出了q-Lévy-Meixner多项式及其母函数统一的显式表示,并且说明当q=0对应自由概率理论的情形,当q↑1时重新得到经典的概率理论情形。 一.Lévy-Meixner正交多项式,母函数与交互作用指数 考虑一维空间。讨论具有有限各阶矩的随机变量X产生的Wick幂,在此论文中,我们总假设X为IR上具有无穷可分分布的随机变量,且具有如下Laplace变换: IE[e^tX]=e^f（t）,（1）其中f在0的某邻域内解析,且f（0）=0,则半不变量,f′（0）=c₁,f″（0）=c₂分别是X的期望和方差。 定义1 如果正交多项式{R_n（x）}的母函数有下面的形式: G（x,t）∶=exp{xα（t）-f（α（t））}=sum from n=0 to ∞ （tⁿ/n（?））P_n（x）,（2）其中α在0的某邻域内解析,α（0）=0,α′（0）=1,f是Laplace指数满足（1）,G（x,t）是正交多项式{P_n（x）}的母函数当且仅当存在在0的某邻域解析的函数g,满足g（0）=0,并使得下式 f（α（t）+α（s））-f（α（t））-f（α（s））=g（st） （3）成立,则我们称函数g为X分布的交互作用指数。更多还原

【Abstract】 In this thesis we mainly study two aspects. The main topic is devoted to Levy-Meixner white noise calculus. First we consider Levy-Meixner polynomials and their generating function over IR and then obtain Levy white noise measure by lifting a 1-dimensional infinitely divisible distribution with finite moments and construct the renormalization kernels （or Wick tensor powers） explicitly in a unified way by lifting the generating function of Meixner orthogonal polynomials. Moreover, we define inner products in n-particle spaces in terms of traces on the "diagonals" and get a unified explicit chaotic representation of Levy-Meixner white noise functionals in terms of interacting Fock spaces. The interacting feature is completely determined by a function g which is referred to as " interaction exponent" and the usual Fock space can be viewed as a quotient space of the interacting Fock space. This method enables us to easily recapture the general form of Levy-Meixner field operators. Next we construct the framework of Levy-Meixner white noise analysis and study the Levy-Meixner field operators and pointwise defined Levy-Meixner field on it. Another topic is devoted to q-deformation of Levy-Meixner Polynomials and their generating function. We obtain a unified explicit form of q-Levy-Meixner Polynomials which is shown to be a reasonable interpolation between free （q = 0） and classical （q = 1） cases.1. Levy-Meixner Polynomials, Generating Function and Interaction ExponentWe first consider one dimensional space and Wick powers of a random variable X with finite moments. In our thesis, we always suppose that X has an infinitely divisible distribution on IR with Laplace transform:where f is analytic in some neighborhood of zero with f（0） = 0. Then the cumulants f’（0） = c₁,f"（0） = c₂ are the expectation and variance of X, respectively. Definition 1 If the generating function of the orthogonal polynomials {P_n（x）} has the following form :where a is analytic in some neighborhood of zero with a（0） = 0 and a’（0） = 1 and / is a Laplace exponent satisfying （1）. Then G（x,t） is a generating function of orthogonal polynomials {Pn（x）} if and only if there exists a function g with g（0） = 0 which is analytic in some neighborhood of zero , such thatf（a（t） + a（s）） - f（a（t）） - f（a（s）） = g（st）. （3）We call the function g interaction exponent of distribution of X .The interaction exponent has the following equivalent definition.Theorem 2 Let X be infinitely divisible with finite moments on FL satisfying （1） and {Pn（x）} be the corresponding orthogonal polynomials having the generating function （2）. Then a function g on IR is the interaction exponent of distribution of X if and only ifn=0oreg（t） =holds where dn is the Hankel determinant of order n of the moment sequence {mn := E[Xn}}.The following theorem gives a characterization of the generating function satisfying （3） and introduces four important parameters Ci,c2,7,/3 which permeate our whole thesis. The proof of the theorem shows that j3 = g"（0）/c2, where g is interaction exponent withAs corollary, the interaction exponent has the following determined forms. Corollary 5 The interaction exponent g of distribution of X only has the following two forms:（1） g（t） = c2t （when /3 = 0）;（2） g（t） = -f ln（l - /3t） （when /3 > 0）. or, write in one form,git) = c2（t + |?2 + ^ + ...）, < t < _<sub>Next we classify the generating functions and the corresponding infinitely divisible distributions according to parameters 7 and /3. Theorem 6 Let G（x,t） satisfy （3）, then（1） if/? = 0,7 = 0, thena（t）=t, f（t）=Clt+jt2,G（x,t） = exp{xt - （cx + jt2）}.In this case, X is Gaussian -/V（ci, C2）.（2） if/3 = 0,7 ^0, thena（t） = -ln（l +7*）, /（*） = （ci - -）*+ %e^ - 1), {-ln（l= exp ■ -.....^ ■’ *’Y ’ ’ ’ ’ * "y T^’ ’ ’ TIn this case, X is a random variable of Poisson type. Especially, if c\ = c2 = A, 7 = 1, thena（t） = ln（l + t）, /（0 = A（e* - 1）,C（a:, t） = exp{a; ln（l + t） - At}.In this case, X is Poisson random variable with parameter A. （3） if/3 >0,72 = 4/3, then2t?, a f 2ta 2c2 2t 4c2 2 i Gix.t） = exp <-----------（ci-------）--------------z-lnf--------） >.v ; 12 + ^ LV -Yy2+^ -y2 l2+7rJiIn this case, X is gamma random variable.（4） if p > 0,72 i- 4,6, and let 1 + jt + fit2 = （1 - at）（l - bt）, then a + b = -7 ,ab = p. So, 1 1 - bt ^ c2 . ae-fe* - 6e"a* / 0 = ci* - -=- In<*（*） =r In 7, / 0 = ci* = In7, a — b 1 — at p a — bG（x, t） = exp \------- In-----------—?— In--------+ nf 2 iN In ^--------{-, \v ’ ’ v\a-b I-at La - b 1 - at f3（a - b） （1 - at）6J JWhen 72 > 4/3, X has a Pascal （negative binomial） distribution; when 72 < 4/3, X is referred to as a Meixner random variable.The following important theorem gives the Szego-Jacobi parameters of {Pn（x）} in terms of Ci,c2,7 and /3. Theorem 7 If {Pn（x）} are orthogonal polynomials satisying （2）, thenPn+1（x） = （x- bn）Pn（x） - wnPn_x（a;）, n E JN, （9）where bn = c\ + 717, wn = ri（e2 + （n — l）/3）.2. Levy White Noise, Moment and Levy-Meixner Renormalization KernelsDenote S（Md） by <S and S*（Md） by <S*, the Schwartz space of testing functions and tempered distributions over JR respectively and denote H := L2（M ）. <Sc, S^ and He are the complexifications of S, S* and H respectively. The dual paring on Sq x 5c is denoted by < ?, ? > which is consistent with the inner product （?, ?） in H and更多还原