### 统计代写|随机分析作业代写stochastic analysis代写|White Noise Schr¨odinger and Heisenberg Equations

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• 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|White Noise Schr¨odinger and Heisenberg Equations

The white noise equations live on spaces of the form
$$H=H_{S} \otimes \Gamma$$

where the Hilbert space $\mathcal{H}_{S}$ is called the initial (or system) space, and the Hilbert space $\Gamma$ is called the noise space.

For 1 -st order white noise equations the typical $\Gamma$ is the same as for Hudson-Parthasarathy equations, i.e. a Fock space over a 1-particle space of the form $L^{2}(\mathbb{R} ; \mathcal{K})$ where $\mathcal{K}$ is another Hilbert space, called the multiplicity space (in mathematics) or polarization space (in physics). A WN Schrödinger (or Hamiltonian) equation is an equation of the form
$$\partial_{t} U_{t}=-i H_{t} U_{t} ; \quad U_{0}=1$$
where $H_{t}=H_{t}^{}$ is a symmetric functional of white noise and the associated Heisenberg equation (from now on we will consider only the inner case) $$\partial_{t} X_{t}=-i\left[H_{t}, X_{t}\right] ; \quad X_{0}=X \in \mathcal{B}(\mathcal{H})$$ Since in the inner case, as explained in Section (7), the solution of the Heisenberg equation has the form $$X_{t}=U_{t} X_{t} U_{t}^{}$$
it will be sufficient to consider the Schrödinger equation.

## 统计代写|随机分析作业代写stochastic analysis代写|Stochastic Equations Associated to 1-st Order WN Schrödinger Equations

The simplest WN equations are the 1 -st order WN Schrödinger equations, for which $H_{t}$ has the form:
$$H_{t}=D b_{t}^{+}+D^{+} b_{t}+T b_{t}^{+} b_{t}+C=D \otimes b_{t}^{+}+\cdots$$
Notice that the right hand side is formally symmetric if
$$T^{+}=T ; \quad C^{+}=C$$
Diffusion WN equations are characterized by the condition:
$$T=0$$
Example.
$$\partial_{t} U_{t}=-i H_{t} U_{t}=-i\left(D b_{t}^{+}+D^{+} b_{t}\right) U_{t}$$
if $D=D^{+}$this becomes
$$\partial_{l} U_{L}=-i H_{l} U_{t}=-i D\left(b_{t}^{+}+b_{l}\right) U_{t}=-i D w_{l} U_{L}$$

in terms of Brownian motion
$$\frac{d}{d t} U_{t}=-i D \frac{d W_{t}}{d t} U_{t}$$
Warning: in Section (4) one might be tempted to use the naive relation
$$\frac{d}{d t} W_{t}=w_{t} \Leftrightarrow d W_{t}=w_{t} d t$$
and to conclude that the classical WN equation (13.2) is equivalent to the classical stochastic differential equation
$$d U_{t}=-i D d W_{t} U_{t}$$
but this would lead to a contradiction because it can be proved that equation (13.4), does not admit any unitary solution while WN Hamiltonian equations of the form (13.1) can be shown to admit unitary solutions.

In fact it is true that WNH equations of the form (13.1) are canonically associated to stochastic differential equations but, for the determination of this stochastic equation, the naive prescription (13.3) is not sufficient and a much subtler rule must be used. The correct answer is given by the following theorem.

Theorem 5. Let $A, C$ and $T=T^{}$ be bounded operators on the initial space $\mathcal{H}{S}$. Then the white noise Schrödinger equation $$\partial{t} U_{t}=-i\left(A b_{t}+A^{} b_{t}^{+}+b_{t}^{+} T b_{t}+C\right) U_{t} ; U_{0}=1$$
$\left(T=T^{} ; C=C^{}\right)$ is equivalent to the following stochastic differential equation
\begin{aligned} d U_{t}=&\left(S D d B_{t}^{+}-D^{*} d B_{t}+\frac{1}{2 \operatorname{Re}\left(\gamma_{-}\right)}(S-1) d N_{t}\right.\ &\left.+\left(-\gamma_{-} D^{+} D+i\left|\gamma_{-}\right|^{2} D^{+} T D-i C\right) d t\right) U_{t} \end{aligned}
where the unitary operator
$$S:=\frac{1-i T}{1+i T}$$
is the Cuyley trursform of T and:
$$D^{+}=i A \frac{1}{1+i T}$$
Remark. The two equations can be interpreted in the weak sense on the total domain of extended number vectors with continuous test functions (the vectors of the form $\xi_{S} \otimes n$ with $\xi_{S} \in \mathcal{H}_{S}$ and $n$ a number vector with continuous test functions).

## 统计代写|随机分析作业代写stochastic analysis代写|The Renormalized Square of Classical WN

We have seen that the quantum decomposition of the 1 -st order classical $\mathrm{WN}$ is:
$$w_{t}=b_{t}^{+}+b_{t}$$
If one tries to do the square of $w_{t}$ naively, one obtains:
$$w_{t}^{2}=\left(b_{t}^{+}+b_{t}\right)^{2}=b_{t}^{+2}+b_{t}^{2}+b_{t}^{+} b_{t}+b_{t} b_{t}^{+}=b_{t}^{+2}+b_{t}^{2}+2 b_{t}^{+} b_{t}+\delta(0)$$

where in the last identity we have applied the commutation relations (3.3) to the case $t=s$. This application is purely formal becanse $\delta(t-s)$ is a distribution and expressions like $\delta(0)$ are meaningless. The standard procedure to overcome this problem is to subtract the diverging quantity $\delta(0)$ (additive renormalization) and to conjecture that the result i.e.:
$$: w_{t}^{2}:=b_{t}^{+2}+b_{t}^{2}+2 b_{t}^{+} b_{t}$$
is, up to a constant, the quantum decomposition of the square of the classical white noise.

However, even after this renormalization the right hand side of (14.2) is ill defined. The problem is that, as will be shown in the following session, expressions like $b_{t}^{+2}, b_{t}^{2}$ are not well defined even as operator valued distributions!

## 统计代写|随机分析作业代写stochastic analysis代写|White Noise Schr¨odinger and Heisenberg Equations

H=H小号⊗Γ

∂吨在吨=−一世H吨在吨;在0=1

## 统计代写|随机分析作业代写stochastic analysis代写|Stochastic Equations Associated to 1-st Order WN Schrödinger Equations

H吨=Db吨++D+b吨+吨b吨+b吨+C=D⊗b吨++⋯

∂吨在吨=−一世H吨在吨=−一世(Db吨++D+b吨)在吨

∂l在大号=−一世Hl在吨=−一世D(b吨++bl)在吨=−一世D在l在大号

dd吨在吨=−一世Dd在吨d吨在吨

dd吨在吨=在吨⇔d在吨=在吨d吨

d在吨=−一世Dd在吨在吨

(吨=吨;C=C)等价于以下随机微分方程
d在吨=(小号Dd乙吨+−D∗d乙吨+12关于⁡(C−)(小号−1)dñ吨 +(−C−D+D+一世|C−|2D+吨D−一世C)d吨)在吨

D+=一世一种11+一世吨

## 统计代写|随机分析作业代写stochastic analysis代写|The Renormalized Square of Classical WN

:在吨2:=b吨+2+b吨2+2b吨+b吨

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