Lévy-Meixner White Noise Calculus

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 â–  -.....^ â– ' *'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 which is consistent with the inner product (?, ?) in H and...