统计代写|随机分析作业代写stochastic analysis代写|Koopman’s Argument and Quantum Extensions

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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|Classical Deterministic Dynamical Systems

The following considerations, due to Koopman, constitute the basis of the algebraic approach to dynamical systems which reduces the study of such systems to the study of 1-parameter groups of unitary operators or of

*-automorphisms of appropriate commutative *-algebras or, at infinitesimal level, to the study of appropriate Schrödinger or Heisenberg equations.
To every ordinary differential equation in $\mathbb{R}^{d}$
$$d x_{t}=b\left(x_{t}\right) d t ; \quad x(0)=x_{0} \in \mathbb{R}^{d}$$
such that the initial value problem admits a unique solution for every initial data $x_{0}$ and for every $t \geq 0$ : one associates the 1-parameter family of maps
$$T_{t}: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$$
characterized by the property that the image of $x_{0}$ under $T_{t}$ is the value of the solution at time $t$ :
$$x_{t}\left(x_{0}\right)=: T_{t} x_{0} ; \quad T_{0}=i d$$
Uniqueness then implies the semigroup property:
$$T_{t} T_{s}=T_{t+s}$$
If the above properties hold not only for every $t \geq 0$, but for every $t \in \mathbb{R}$, then the system is called reversible. In this case each $T_{t}$ is invertible and
$$T_{t}^{-1}=T_{-t}$$
Typical examples of these systems are the classical Hamiltonian systems. They have the additional property that the maps $T_{t}$ preserve the Lebesgue measure (Liouville’s theorem).

Abstracting the above notion to an arbitrary measure space leads to the notion of (deterministic) dynamical system:

Definition 3. Let $(S, \mu)$ be a measure space. A classical, reversible, deterministic dynamical system is a pair:
$$\left{(S, \mu) ;\left(T_{t}\right) t \in \mathbb{R}\right}$$
where $T_{t}: S \rightarrow S$ ( $\left.t \in \mathbb{R}\right)$ is a 1 -parameter group of invertible bi-measurable maps of $(S, \mu)$ admitting $\mu$ as a quasi-invariant measure:
$$\mu \circ T_{t} \sim \mu$$
The quasi-invariance of $(S, \mu)$ is equivalent to the existence of a $\mu$-almost everywhere invertible Radon-Nikodym derivative:
$$\begin{gathered} \frac{d\left(\mu \circ T_{t}\right)}{d \mu}=: p_{\mu, t} \in L^{1}(S, \mu) \ p_{\mu, t}>0 ; \mu-\text { a.e.; } \quad \int_{S} p_{\mu, t}(s) d \mu(s)=1 \end{gathered}$$

统计代写|随机分析作业代写stochastic analysis代写|Stochastic Extension of Koopman’s Approach

In the present section we will replace, in the above Koopman’s argument, the deterministic trajectory $\left(x_{t}\left(x_{0}\right)\right)$ by a stochastic process $\left(X_{t}\right)$ and show how the general algebraization procedure described in Section (8), when applied to the simple and important example of a classical diffusion flow $\left(X_{t}\right)$, naturally leads to a classical stochastic generalization of the Heisenberg equation.

Let $\left(X_{t}\right)$ denote the real valued solution of the classical stochastic differential equation
$$d X_{t}=l d t+a d W_{t} ; X(0)=X_{0}$$
driven by classical Brownian motion $\left(W_{t}\right)$ and with adapted coefficients $l, a$ which guarantee the existence and uniqueness of a strong solution for all initial data $X_{0}$ in $L^{2}(\mathbb{R})$ and for all times. The initial value $X_{0}$ is a random variable independent of $\left(W_{t}\right)$. By Itô’s formula equation (10.1) is equivalent to
$$d f\left(X_{t}\right)=\left(l \partial_{x} f+\frac{1}{2} a^{2} \partial_{x}^{2} f\right) d t+a \partial_{x} f d W_{t}$$
where $f: \mathbb{R} \rightarrow \mathbb{R}$ varies in a space of sufficiently smooth functions.
Since $X_{t}$ depends also on the initial condition $x \in \mathbb{R}, f\left(X_{t}\right)$ is realized as multiplication operator on
$$L^{2}(\mathbb{R}) \otimes L^{2}(\Omega, \mathcal{F}, P)$$
where $(\Omega, \mathcal{F}, P)$ is the probability space of the increment process of the Brownian motion. In the following we shall simply write $f\left(\right.$ or $f\left(X_{t}\right)$ ) to mean the multiplication operator by $f\left(\right.$ or $\left.f\left(X_{t}\right)\right)$. When confusion can arise we shall write $M_{f}$ or $M_{f\left(X_{t}\right)}$. With these notations one has:
$$\left[\partial_{x}, f\right]=\left[\partial_{x}, M_{f}\right]=\partial_{x} \cdot f-f \cdot \partial_{x}=\partial_{x} f=M_{\partial_{x} f}$$
Therefore
$$\left[\partial_{x},\left[\partial_{x}, f\right]\right]=\left[\partial_{x},\left[\partial_{x}, M_{f}\right]\right]=M_{\partial_{x}^{2} f}=\Delta f=M_{\Delta f}$$
Introducing the momentum operator on $L^{2}(\mathbb{R})$ :
$$p:=\frac{1}{i} \partial_{x}$$
defined on those functions in $L^{2}(\mathbb{R})$ with a derivative also in $L^{2}(\mathbb{R})$, we can write
$$\partial_{x} f=i[p, f] ; \quad \partial_{x}^{2} f=-[p,[p, f]]$$
More generally, interpreting both $f$ and $l$ as multiplication operators and using the fact that $f$ commutes with $l$, one finds:
\begin{aligned} l f^{\prime}=& l \partial_{x} f=l i[p, f]=\frac{i}{2} l[p, f]+\frac{i}{2}[p, f] l=\frac{i}{2} l p f \ &-\frac{i}{2} l f p+\frac{i}{2} p f l-\frac{i}{2} f p l=i\left[\frac{1}{2} l p+\frac{1}{2} p l, f\right]=: i[p(l), f] \end{aligned}
and therefore:
$$a^{2} \partial_{x}^{2} f=a \partial_{x} a \partial_{x} f-a\left(\partial_{x} a\right) \partial_{x} f=-\left[p(a),[p(a), f]-i\left[p\left(a \partial_{x} a\right), f\right]\right.$$

统计代写|随机分析作业代写stochastic analysis代写|Quantum Stochastic Schrödinger and Heisenberg Equations

The transition from classical to quantum stochastic Schrödinger and Heisenberg equations is now accomplished by using the quantum decomposition of the classical Brownian motion $d W_{t}=d B_{t}^{+}+d B_{t}$ and allowing for different coefficients of the quantum stochastic differentials $d B_{t}^{+}$and $d B_{t}$.

Differentiating the unitarity conditions for $U_{t}$ and using the HudsonParthasarathy Itô table, we deduce a relation between the coefficients of $d B_{t}^{+}$, $d B_{t}$ and $d t$. The final form of the equation is then:
$$d U_{t}=\left(D d B_{t}^{+}-D^{+} d B_{t}-\left[\frac{1}{2} D^{+} D+i H\right] d t\right) U_{t}$$
where $D$ and $H$ are arbitrary, say bounded, operators and $H=H^{}$. The same argument, applied to a more general equation, including also the number differential $d N_{t}$ leads to the most general Hudson-Parthasarathy stochastic Schrödinger equation: $$d U_{t}=\left(S D d B_{t}^{+}-D^{} d B_{t}+(S-1) d N_{t}+\left(-\frac{1}{2} D^{+} D+i H\right) d t\right) U_{t}$$
where $D$ and $H$ are as above and $S$ must be a unitary operator. Notice that, contrary to the diffusion case (11.1), here the Hamiltonian nature of the equation is lost even at the level of the martingale term: the non Hamiltonian nature of equation (11.2) is not due only to the presence of the dissipative term $D^{+} D / 2$ but also of the unitary operator $S$. The deep meaning of this apparently strange structure can only be understood in terms of quantum white noise calculus (see Section (13) below).

统计代写|随机分析作业代写stochastic analysis代写|Classical Deterministic Dynamical Systems

dX吨=b(X吨)d吨;X(0)=X0∈Rd

X吨(X0)=:吨吨X0;吨0=一世d

\left{(S, \mu) ;\left(T_{t}\right) t \in \mathbb{R}\right}\left{(S, \mu) ;\left(T_{t}\right) t \in \mathbb{R}\right}

μ∘吨吨∼μ

d(μ∘吨吨)dμ=:pμ,吨∈大号1(小号,μ) pμ,吨>0;μ− ae; ∫小号pμ,吨(s)dμ(s)=1

统计代写|随机分析作业代写stochastic analysis代写|Stochastic Extension of Koopman’s Approach

dX吨=ld吨+一种d在吨;X(0)=X0

dF(X吨)=(l∂XF+12一种2∂X2F)d吨+一种∂XFd在吨

[∂X,F]=[∂X,米F]=∂X⋅F−F⋅∂X=∂XF=米∂XF

[∂X,[∂X,F]]=[∂X,[∂X,米F]]=米∂X2F=ΔF=米ΔF

p:=1一世∂X

∂XF=一世[p,F];∂X2F=−[p,[p,F]]

lF′=l∂XF=l一世[p,F]=一世2l[p,F]+一世2[p,F]l=一世2lpF −一世2lFp+一世2pFl−一世2Fpl=一世[12lp+12pl,F]=:一世[p(l),F]

统计代写|随机分析作业代写stochastic analysis代写|Quantum Stochastic Schrödinger and Heisenberg Equations

d在吨=(Dd乙吨+−D+d乙吨−[12D+D+一世H]d吨)在吨

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