### 统计代写|随机分析作业代写stochastic analysis代写|Theory and Applications of Infinite

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## 统计代写|随机分析作业代写stochastic analysis代写|Dimensional Oscillatory Integrals

Professor K. Itò’s work on the topic of infinite dimensional oscillatory integrals has been very germinal and stimulated much of the subsequent research in this area. It is therefore a special honour and pleasure to be able to dedicate the present pages to him. We shall give a short exposition of the theory of a particular class of functionals, the oscillatory integrals:
$$I^{\text {ᄒ}}(f)=\quad ” \int_{\Gamma} e^{i \frac{\psi}{*}(\gamma)} f(\gamma) d \gamma “$$
where $\Gamma$ denotes either a finite dimensional space (e.g. $\mathbb{R}^{s}$, or an s-dimensional differential manifold $M^{s}$ ), or an infinite dimensional space (e.g. a “path space”). $\Phi: \Gamma \rightarrow \mathbb{R}$ is called phase function, while $f: \Gamma \rightarrow \mathbb{C}$ is the function to be integrated and $\epsilon \in \mathbb{R} \backslash{0}$ is a parameter. The symbol $d \gamma$ denotes a “flat” measure. In particular, if $\operatorname{dim}(\Gamma)<\infty$ then $d \gamma$ is the Riemann-Lebesgue volume measure, while if $\operatorname{dim}(\Gamma)=\infty$ an analogue of Riemann-Lebesgue measure is not mathematically defined and $d \gamma$ is just a heuristic expression.

## 统计代写|随机分析作业代写stochastic analysis代写|Finite Dimensional Oscillatory Integrals

In the case where $\Gamma$ is a finite dimensional vector space, i.e. $\Gamma=\mathbb{R}^{s}, s \in \mathbb{N}$, the expression (1.1)
$$” \int_{\mathbb{R}^{}} e^{i \frac{\text { s্ }}{\varepsilon}(\gamma)} f(\gamma) d \gamma ”$$ can be defined as an improper Riemann integral. The study of finite dimensional oscillatory integrals of the type (1.2) is a classical topic, largely developed in connection with several applications in mathematics (such as the theory of Fourier integral operators $[48]$ ) and physics. Interesting examples of integrals of the form (1.2) in the case $s=1, \epsilon=1, f=\chi[0, w], w>0$, and $\Phi(x)=\frac{\pi}{2} x^{2}$, are the Fresnel integrals, that are applied in optics and in the theory of wave diffraction. If $\Phi(x)=x^{3}+a x, a \in \mathbb{R}$ we obtain the Airy integrals, introduced in 1838 in connection with the theory of the rainbow. Particular interest has been devoted to the study of the asymptotic behavior of integrals (1.2) when $\epsilon$ is regarded as a small parameter converging to 0 . Originally introduced by Stokes and Kelvin and successively developed by several mathematicians, in particular van der Corput, the “stationary phase method” provides a powerful tool to handle the asymptotics of (1.2) as $\epsilon \downarrow 0$. According to it, the main contribution to the asymptotic behavior of the integral should come from those points $\gamma \in \mathbb{R}^{}$ which belong to the critical manifold:
$$\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}$$
that is the points which make stationary the phase function $\Phi$. Beautiful mathematical work on oscillatory integrals and the method of stationary phase is connected with the mathematical classification of singularities of algebraic and geometric structures (Coxeter indices, catastrophe theory), see, e.g. [31].

## 统计代写|随机分析作业代写stochastic analysis代写|Infinite Dimensional Oscillatory Integrals

The extension of the results valid for $\Gamma=\mathbb{R}^{s}$ to the case where $\Gamma$ is an infinite dimensional space is not trivial. The main motivation is the study of the “Feynman path integrals”, a class of (heuristic) functional integrals introduced by R.P. Feynman in $1942^{1}$ in order to propose an alternative, Lagrangian, formulation of quantum mechanics. According to Feynman, the solution of the Schrödinger equation describing the time evolution of the state $\psi \in L^{2}\left(\mathbb{R}^{d}\right)$ of a quantum particle moying in a potential $V$
$$\left{\begin{array}{l} i \hbar \frac{\partial}{\partial t} \psi=-\frac{n^{2}}{2 m} \Delta \psi+V \psi \ \psi(0, x)=\psi_{0}(x) \end{array}\right.$$

(where $m>0$ is the mass of the particle, $\hbar$ is the reduced Planck constant, $t \geq 0, x \in \mathbb{R}^{d}$ ) can be represented by a “sum over all possible histories”, that is an integral over the space of paths $\gamma$ with fixed end point
$$\vartheta \gamma^{\prime}(t, x)=-\int_{{\gamma \mid \gamma(t)=x}} e^{\hbar S_{t}(\gamma)} \gamma_{\gamma}(\gamma(0)) d \gamma^{\eta}$$
$S_{t}(\gamma)=S^{0}(\gamma)-\int_{0}^{t} V(s, \gamma(s)) d s, S^{0}(\gamma)=\frac{m}{2} \int_{0}^{t}|\dot{\gamma}(s)|^{2} d s$, is the classical action of the system evaluated along the path $\gamma$ and $d \gamma$ a heuristic “flat” measure on the space of paths (see e.g. [40] for a physical discussion of Feynman’s approach and its applications). The Feynman path integrals (1.4) can be regarded as oscillatory integrals of the form (1.1), where
$$\Gamma=\left{\text { paths } \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\right}$$
the phase function $\Phi$ is the classical action functional $S_{t}, f(\gamma)=\psi_{0}(\gamma(0))$, the parameter $\epsilon$ is the reduced Planck constant $\hbar$ and $d \gamma$ denotes heuristically
$$d \gamma={ }^{\alpha} C \prod_{s \in[0, t]} d \gamma(s)^{“},$$
$C:=”\left(\int_{{\gamma \mid \gamma(t)=x}} e^{\frac{1}{\hbar} S_{0}(\gamma)} d \gamma\right)^{-1 “}$ being a normalization constant
The Feynman’s path integral representation (1.4) for the solution of the Schrödinger equation is particularly suggestive. Indeed it creates a connection between the classical (Lagrangian) description of the physical world and the quantum one and makes intuitive the study of the semiclassical limit of quantum mechanics, that is the study of the detailed behavior of the wave function $\psi$ in the case where the Planck constant $\hbar$ is regarded as a small parameter. According to an (heuristic) application of the stationary phase method, in the limit $\hbar \downarrow 0$ the main contribution to the integral (1.4) should come from those paths $\gamma$ which make stationary the action functional $S_{t}$. These, by Hamilton’s least action principle, are exactly the classical orbits of the system.

Despite its powerful physical applications, formula (1.4) lacks mathematical rigour, in particular the “flat” measure $d \gamma$ given by (1.5) has no mathematical meaning.

## 统计代写|随机分析作业代写stochastic analysis代写|Dimensional Oscillatory Integrals

K. Itò 教授关于无限维振荡积分的研究非常具有开创性，并激发了该领域的许多后续研究。因此，能够将本页献给他是一种特殊的荣幸和荣幸。我们将对一类特殊泛函的理论进行简短的阐述，即振荡积分：
ᄒ一世ᄒ(F)=”∫Γ和一世ψ∗(C)F(C)dC“

## 统计代写|随机分析作业代写stochastic analysis代写|Finite Dimensional Oscillatory Integrals

্”∫R和一世 s ্ e(C)F(C)dC”可以定义为不正确的黎曼积分。(1.2) 类型的有限维振荡积分的研究是一个经典课题，主要与数学中的几种应用（例如傅里叶积分算子理论[48]) 和物理学。本例中 (1.2) 形式的积分的有趣示例s=1,ε=1,F=χ[0,在],在>0， 和披(X)=圆周率2X2, 是菲涅耳积分，应用于光学和波衍射理论。如果披(X)=X3+一种X,一种∈R我们获得了 1838 年与彩虹理论相关的艾里积分。特别感兴趣的是积分（1.2）的渐近行为的研究，当ε被认为是一个收敛到 0 的小参数。最初由 Stokes 和 Kelvin 提出并由几位数学家，特别是 van der Corput 相继开发，“平稳相法”提供了一个强大的工具来处理 (1.2) 的渐近性：ε↓0. 据此，对积分渐近行为的主要贡献应该来自这些点C∈R属于临界流形：
\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}

## 统计代写|随机分析作业代写stochastic analysis代写|Infinite Dimensional Oscillatory Integrals

$$\左{一世⁇∂∂吨ψ=−n22米Δψ+在ψ ψ(0,X)=ψ0(X)\对。$$

（在哪里米>0是粒子的质量，⁇是简化的普朗克常数，吨≥0,X∈Rd) 可以表示为“所有可能历史的总和”，即路径空间上的积分C带固定端点
ϑC′(吨,X)=−∫C∣C(吨)=X和⁇小号吨(C)CC(C(0))dC这

\Gamma=\left{\text { 路径} \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\对}\Gamma=\left{\text { 路径} \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\对}

dC=一种C∏s∈[0,吨]dC(s)“,
C:=”(∫C∣C(吨)=X和1⁇小号0(C)dC)−1“作为归一化常数

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