### 统计代写|随机分析作业代写stochastic analysis代写|No–go Theorems

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|No–go Theorems

The first no-go theorem, showing that it is not true that, if a Lie algebra admits a Fock representation, then any associated current algebra also admits one was proved by Śniady [Śnia99]. In the terminology intruduced in the present paper Śniady’s result can be rephrased as follows:

Theorem 10. The Schrödinger algebra admits a Fock representation but its associated current algebra over $\mathbb{R}$ with Lebesgue measure doesn’t.

Since the Schrödinger algebra is contained in the full oscillator algebra, which clearly admits a Fock representation, Sniady’s theorem also rules out the possibility of a Fock representation for the current algebra of the full oscillator algebra over $\mathbb{R}$ with Lebesgue measure.

Recalling, from the examples at the end of Section (18), that the Schrödinger algebra is the smallest *-Lie algebra containing the oscillator algebra (with generators $\left{a^{+}, a, a^{+} a, 1\right}$ ) and the square-oscillator algebra, i.e. $s l(2, \mathbb{R}$ ) (with generators $\left{a^{+2}, a^{2}, a^{+} a, 1\right}$ ), we see that the difficulty comes from the combination of two closed Lie algebras. More precisely: consider the two sets of generators
$$\begin{gathered} \left{a^{+}, a, a^{+} a, 1\right} \ \left{a^{+2}, a^{2}, a^{+} a, 1\right} \end{gathered}$$
We know that the current algebra over $\mathbb{R}^{d}$ associated to each of them has a Fock representation. However the union of the two sets, i.e.
$$\left{a^{+}, a, a^{+2}, a^{2}, a^{+} a, 1\right}$$
is also a set of generators of a *-Lie algebra whose associated current algebra over $\mathbb{R}^{d}$ does not admit a Fock representation.

Notice that the first of the two algebras is generated by the first powers of the white noise and the number operator while the second one is generated by the second powers of the white noise and the number operator. An extrapolation of this argument suggested the hope that a similar thing could happen also for the higher powers, i.e. that, denoting $\mathcal{G}{3}$ the -Lie algebra generated by the cube of the white noise $b{t}^{3}$ and the number operator; and, for $n \geq 4, \mathcal{G}{n}$ the $$-Lie algebra generated by the number operator and the smallest power of the white noise not included in \bigcup{1 \leq k \leq n-1} \mathcal{G}{k}, the current algebra of \mathcal{G}{n} over \mathbb{R}^{d} admits a Fock representation. This hope was ruled out by the following generalization of Sniady’s theorem, due to Accardi, Boukas and Franz [AcBouFr05] and by its corollary reported below. ## 统计代写|随机分析作业代写stochastic analysis代写|Connection with an Old Open Problem in Classical Probability Since the vacuum distribution of the first order classical white noise is a Gaussian, any reasonable renormalization should lead to the conclusion that the n-th power of the first order classical white noise is still the n-th power of a Gaussian. But the \delta-correlation implies that the corresponding integrated process will be a stationary additive independent increment process on \mathbb{R}. These heuristic ideas, which can be put in a satisfactory mathematical form with some additional work, lead to the conjecture that a necessary condition for the existence of the n-th power of white noise, renormalized as in [AcBouFr05], is that the n-th power of a classical Gaussian random variable is infinitely divisible. The n-th powers of the standard Gaussian random variable \gamma and their distributions have been widely studied. It is known that, \forall k \geq 1 \gamma^{2 k} is infinitely divisible, but it is not known if, \forall k \geq 1 \gamma^{2 k+1} is infinitely divisible (and the experts conjecture that, at least for \gamma^{3}, the answer is negative). ## 统计代写|随机分析作业代写stochastic analysis代写|Renormalized Powers of White Noise and the Virasoro-Zamolodchikov Algebra In the present section we will use the notations of Section (20) and the results of the papers [AcBou06a, AcBou06b, AcBou06c] which contain the proofs of all the results discussed here. The formal extension of the white noise commutation relations to the associative *-algebra generated by b_{t}, b_{s}^{\dagger}, 1, called from now on the renormalized higher powers of (Boson) white noise (RHPWN) algebra, leads to the identities:$$
\begin{aligned}
{\left[b_{t}^{\dagger^{n}} b_{t}^{k}, b_{s}^{\dagger} b_{s}^{K}\right]=} & \epsilon_{k, 0} \epsilon_{N, 0} \sum_{L \geq 1}\left(\begin{array}{c}
k \
L
\end{array}\right) N^{(L)} b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-L}} b_{t}^{k-L} b_{s}^{K} \delta^{L}(t-s) \
&-\epsilon_{K, 0} \epsilon_{n, 0} \sum_{L \geq 1}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)} b_{s}^{\dagger} b_{t}^{\dagger^{n-L}} b_{s}^{K-L} b_{t}^{k} \delta^{L}(t-s)
\end{aligned}
$$In Section (20) we have given a meaning to these formal commutation relations, i.e. to the ill defined powers of the \delta-function, through the renormalization prescription (20.2). In the present note we will use a different renormalization rule, introduced in [AcBou06a] and whose motivations are discussed in [AcBou06b, AcBou06c], namely:$$
$$where the right hand side is defined as a convolution of distributions. Using this (23.1) can be rewritten in the form:$$
\begin{aligned}
{\left[b_{t}^{\dagger^{n}} b_{t}^{k}, b_{s}^{\dagger^{N}} b_{s}^{K}\right]=} & \epsilon_{k, 0} \epsilon_{N, 0}\left(k N b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-1}} b_{t}^{k-1} b_{s}^{K} \delta(t-s)\right.\
&\left.+\sum_{L \geq 2}\left(\begin{array}{l}
k \
L
\end{array}\right) N^{(L)} b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-L}} b_{t}^{k-L} b_{s}^{K} \delta(s) \delta(t-s)\right) \
&-\epsilon_{K, 0} \epsilon_{n, 0}\left(K n b_{s}^{\dagger^{N}} b_{t}^{\dagger^{\dagger-1}} b_{s}^{K-1} b_{t}^{k} \delta(t-s)\right.\
&\left.+\sum_{L \geq 2}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)} b_{s}^{\dagger^{N}} b_{t}^{\dagger^{n-L}} b_{s}^{K-L} b_{t}^{k} \delta(s) \delta(t-s)\right)
\end{aligned}
$$Introducing test functions and the associated smeared fields$$
B_{k}^{n}(f):=\int_{\mathbb{R}} f(t) b_{t}^{\dagger^{n}} b_{t}^{k} d t
$$The commutation relations (23.2) become:$$
\begin{aligned}
&{\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f)\right]=\left(\epsilon_{k, 0} \epsilon_{N, 0} k N-\epsilon_{K, 0} \epsilon_{n, 0} K n\right) B_{K+k-1}^{N+n-1}(\bar{g} f)} \
&\quad+\sum \sum_{L=2}^{(K \wedge n) \vee(k \wedge N)} \theta_{L}(n, k ; N, K) \bar{g}(0) f(0) b_{0}^{\dagger^{N+n-l}} b_{0}^{K+k-I} \
&\theta_{L}(N, K, n, k) \cdot-\varepsilon_{K, 0} \varepsilon_{n, 0}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)}-\tau_{k, 0} \kappa_{N, 0}\left(\begin{array}{c}
k \
L
\end{array}\right) N^{(L)}
\end{aligned}
$$The commutation relations (23.4) still contain the ill defined symbol b_{0}^{\dagger^{N+n-1}} b^{K+k-l}. However, if the test function space is chosen so that$$
f(0)=g(0)=0
$$then the singular term in (23.4) vanishes and the commutation relations (23.4) become:$$
\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f)\right]{R}:=(k N-K n) B{k+K-1}^{n+N-1}(\bar{g} f)

## 统计代写|随机分析作业代写stochastic analysis代写|Renormalized Powers of White Noise and the Virasoro-Zamolodchikov Algebra

[b吨†nb吨ķ,bs†bsķ]=εķ,0εñ,0∑大号≥1(ķ 大号)ñ(大号)b吨†nbs†ñ−大号b吨ķ−大号bsķd大号(吨−s) −εķ,0εn,0∑大号≥1(ķ 大号)n(大号)bs†b吨†n−大号bsķ−大号b吨ķd大号(吨−s)

dl(吨−s)=d(s)d(吨−s),l=2,3,4,…

[b吨†nb吨ķ,bs†ñbsķ]=εķ,0εñ,0(ķñb吨†nbs†ñ−1b吨ķ−1bsķd(吨−s) +∑大号≥2(ķ 大号)ñ(大号)b吨†nbs†ñ−大号b吨ķ−大号bsķd(s)d(吨−s)) −εķ,0εn,0(ķnbs†ñb吨††−1bsķ−1b吨ķd(吨−s) +∑大号≥2(ķ 大号)n(大号)bs†ñb吨†n−大号bsķ−大号b吨ķd(s)d(吨−s))

[乙ķn(G¯),乙ķñ(F)]=(εķ,0εñ,0ķñ−εķ,0εn,0ķn)乙ķ+ķ−1ñ+n−1(G¯F) +∑∑大号=2(ķ∧n)∨(ķ∧ñ)θ大号(n,ķ;ñ,ķ)G¯(0)F(0)b0†ñ+n−lb0ķ+ķ−一世 θ大号(ñ,ķ,n,ķ)⋅−eķ,0en,0(ķ 大号)n(大号)−τķ,0ķñ,0(ķ 大号)ñ(大号)

F(0)=G(0)=0

$$\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f) \right] {R}:=(k NK n) B {k+K-1}^{n+N-1}(\bar{g} f)$$

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