### 统计代写|随机分析作业代写stochastic analysis代写|Basic New Idea: Renormalize the Commutation Relations

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机分析作业代写stochastic analysis代写|Renormalize the Commutation Relations

The problem of giving a meaning to expressions like $b_{t}^{2}, b_{t}^{+2}$ has its origins in the fact that the commutation relations
$$\left[b_{s}, b_{t}^{+}\right]=\delta(t-s)$$
imply that
$$\left[b_{s}^{2}, b_{t}^{+2}\right]=4 \delta(t-s) b_{s}^{+} b_{t}+2 \delta(t-s)^{2}$$
But what does it mean $\delta(t-s)^{2}$ ? We found in the literature [Ivanov79] (see also [BogLogTod69, Vlad66]) the following prescription: On an appropriate test function space the following identity holds
$$\delta(t)^{2}=c \delta(t)$$
where the constant $c \in \mathbb{C}$ is arbitrary. (A poof of this statement and the description of the test function space can be found in [AcLuVo99].)

Using this prescription in (15.1) we obtain the renormalized commutation relations:
$$\left[b_{s}^{2}, b_{t}^{+2}\right]=4 \delta(t-s) b_{s}^{+} b_{t}+2 c \delta(t-s)$$
Moreover (without any renormalization!)
$$\left[b_{s}^{2}, b_{t}^{+} b_{t}\right]=2 \delta(t-s) b_{t}^{2}$$
From (15.2) and (15.3) it follows that, after renormalization, the self-adjoint set of operators
$$b_{s}^{2}, b_{s}^{+2}, b_{t}^{+} b_{t}, c=\text { (central element) }$$
is closed under commutators, i.e. the linear span of these operators is a *-Lie algebra.

## 统计代写|随机分析作业代写stochastic analysis代写|Existence of Fock Representations

Having defined the Fock representation the first problem is its existence. In case of the first order white noise this is a well known result since the early days of quantum theory.

Theorem (Fock 1930). The Fock representation of the first order white noise (i.e. the current algebra over $\mathbb{R}$ of the CCR. Lie algebra $\left[a, a^{+}\right]=1$, for the notion of current algebra see Section (18)) exists and is unique up to unitary isomorphism.

The analogue for the RSWN Lie algebra was established more recently. Theorem (Accardi, Lu, Volovich 1999). The Fock representation of the second order white noise (current algebra over $\mathbb{R}$ of the Lie algebra $s l(2, \mathbb{R})$ ) exists and is unique up to unitary isomorphism.

A direct proof of this result is a nontrivial application of the principle that algebra implies statistics, described in its simplest form in Section (3): one proves that, if the required Fock representation exists, then the scalar product of two number vectors is uniquely determined by the commutation relations (15.4), (15.5), (15.6), (15.7) and the Fock property (15.8). Then, and this is the difficult part, one has to prove that this is indeed a scalar product, i.e. that it is positive definite (cf. [AcLuVo99]).

In Section (21) we will come back to this point. Before that let us analyze some consequences of the above theorem. More precisely let us apply to this case the basic general principle of QP discussed in Section (3): algebra implies statistics. In Section (3) we have seen that the application of this principle to the first order white noise shows that the corresponding algebra implies Gaussian and Poisson statistics. It is therefore natural to rise the following question:

Which statistics is implied by the algebra of the renormalized Square of $W N$ ?

The answer to this question was given by Accardi, Franz and Skeide in the paper [AcFrSk00].

## 统计代写|随机分析作业代写stochastic analysis代写|Classical Subprocesses Associated to the Second Order White Noise

To understand this answer it is convenient to take as starting point the analogy with the q-decomposition of the compensated classical Poisson process with intensity $\beta^{-1}$
$$\dot{p}{t}=b{t}^{+}+b_{t}+\beta b_{t}^{+} b_{t}$$
At the end of Section (5) we have seen that $\beta=0$ is the only critical case and corresponds to the transition from classical scalar valued standard compensated Poisson process with intensity $\beta^{-1}$.

This analysis is extended in the paper [AcFrSk00] to the renormalized square of white noise by considering the classical subprocesses
$$X_{\beta}(t):=b_{t}^{+2}+b_{t}^{2}+\beta b_{t}^{+} b_{t}$$
where $\beta$ is a real number. It is then proved that now there are 2 critical values of $\beta$, namely:
$$\beta=\pm 2$$
The value $+2$ corresponding to the renormalized square of the position (classical) white noise, i.e.

\begin{aligned} w_{t}^{2} &=\left|b_{t}^{+}+b_{t}\right|^{2}=b_{t}^{+2}+b_{t}^{2}+b_{t}^{+} b_{t}+b_{t} b_{t}^{+} \ &=b_{t}^{+2}+b_{t}^{2}+2 b_{t}^{+} b_{t}+\delta(0) \equiv b_{t}^{+2}+b_{t}^{2}+2 b_{t}^{+} b_{t} \end{aligned}
and the value $-2$ to the renormalized square of the momentum white noise, i.e.
$$\left(b_{t}^{+}-b_{t}\right) / i$$
The vacuum distribution of both processes is the Gamma-process
$$\mu(d x)=\frac{|x|^{m_{0}-1}}{\Gamma\left(m_{0}\right)} e^{-\beta x} \chi_{\beta \mathbb{R}{+}}$$ whose parameter $m{0}>0$ is uniquely determined by the choice of the unitary representation of $S L(2, \mathbb{R})$ corresponding to the representation of the SWN algebra (cf. [ACFRSK00]).

In this functional realization the number vectors become the Laguerre polynomials which are orthogonal for the Gamma distribution.

Since the Gamma-distributions are precisely the distributions of the $\chi^{2}$ random variables, this result confirms the naive intuition that the distribution of the [renormalized] square of white noise should be a Gamma-distributions.
For $|\beta|<2$ the intensity of the jumps is not strong enough and each of the classical random variables
$$X_{\beta}(t):=b_{t}^{+2}+b_{t}^{2}+\beta b_{t}^{+} b_{t}$$
still has a density whose explicit form, in terms of the $\Gamma$-function is:
$$\mu(d x)=C \exp \left(-\frac{(2 \arccos \beta+\pi) x}{2 \sqrt{1-\beta^{2}}}\right)\left|\Gamma\left(\frac{m_{0}}{2}+\frac{i x}{2 \sqrt{1-\beta^{2}}}\right)\right|^{2}$$

## 统计代写|随机分析作业代写stochastic analysis代写|Renormalize the Commutation Relations

[bs,b吨+]=d(吨−s)

[bs2,b吨+2]=4d(吨−s)bs+b吨+2d(吨−s)2

d(吨)2=Cd(吨)

[bs2,b吨+2]=4d(吨−s)bs+b吨+2Cd(吨−s)

[bs2,b吨+b吨]=2d(吨−s)b吨2

bs2,bs+2,b吨+b吨,C= （中心元素）

## 统计代写|随机分析作业代写stochastic analysis代写|Existence of Fock Representations

RSWN 李代数的类比是最近建立的。定理 (Accardi, Lu, Volovich 1999)。二阶白噪声的 Fock 表示（电流代数超过R李代数的sl(2,R)) 存在并且在单一同构之前是唯一的。

Accardi、Franz 和 Skeide 在论文 [AcFrSk00] 中给出了这个问题的答案。

## 统计代写|随机分析作业代写stochastic analysis代写|Classical Subprocesses Associated to the Second Order White Noise

p˙吨=b吨++b吨+bb吨+b吨

Xb(吨):=b吨+2+b吨2+bb吨+b吨

b=±2

(b吨+−b吨)/一世

$$\mu(dx)=\frac{|x|^{m_{0}-1}}{\Gamma\left(m_{0}\right)} e ^{-\beta x} \chi_{\beta \mathbb{R} {+}}$$ 其参数 $m {0}>0一世s在n一世q在和l是d和吨和r米一世n和db是吨H和CH这一世C和这F吨H和在n一世吨一种r是r和pr和s和n吨一种吨一世这n这FSL(2, \mathbb{R})$ 对应于 SWN 代数的表示（参见 [ACFRSK00]）。

Xb(吨):=b吨+2+b吨2+bb吨+b吨

μ(dX)=C经验⁡(−(2阿尔科斯⁡b+圆周率)X21−b2)|Γ(米02+一世X21−b2)|2

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。