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

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## 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

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