Posted on Categories:数学代写, 随机分析

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数学代写|随机分析代写Stochastic Calculus代考|The Diffeomorphism Group Example

If $M$ is a compact smooth manifold and $X$ is smooth we may consider an equation on the space of smooth diffeomorphisms $\operatorname{Diff}(M)$. Define $\tilde{X}(f)(x)=X(f(x))$ and $\tilde{X}0(f)(x)=X_0(f(x))$ and consider the $\operatorname{SDE}$ on $\operatorname{Diff}(M)$ : $$\mathrm{d} f_t=\tilde{X}\left(f_t\right) \circ \mathrm{d} B_t+\tilde{X}_0\left(f_t\right) \mathrm{d} t$$ with $f_0(x)=x$. Then, $f_t(x)$ is solution to $\mathrm{d} x_t=X\left(x_t\right) \circ \mathrm{d} B_t$ with initial point $x$. Fix $x_0 \in M$, we have a map $\theta: \operatorname{Diff}(M) \rightarrow M$ given by $\theta(f)=f\left(x_0\right)$. Let $\mathcal{B}=\frac{1}{2} L{\tilde{X}i} L{\tilde{X}i}$ and $\mathcal{A}=\frac{1}{2} L{X_i} L_{X_i}$. Then,
$$h_f(v)(x)=\tilde{X}(f)\left(Y\left(f\left(x_0\right)\right) v\right)(x)=X(f(x))\left(Y\left(f\left(x_0\right)\right) v\right) .$$

Consider the polar coordinates in $\mathbf{R}^n$, with the origin removed. Consider the conditional expectation of a Brownian motion $W_t$ on $\mathbf{R}^n$ on $\left|W_t\right|$ where $\left|W_t\right|$, and $n$-dimensional Bessel Process, $n>1$, lives in $\mathbf{R}_{+}$. For $n=2$, we are in the situation that $p: \mathbf{R}^2 \rightarrow \mathbf{R}$ given by $p:(r, \theta) \mapsto r$. The $\mathcal{B}$ and $\mathcal{A}$ diffusion are the Laplacians, $\mathcal{A}^H=\frac{\partial^2}{\partial r^2}$. The map $p(r, \theta)=r^2$ would result the lifting map $v \frac{\partial}{\partial x} \mapsto\left(\frac{v}{2 r}, 0\right)=\frac{v}{2 r} \frac{\partial}{\partial r}$

At this stage, we note that if $B_t$ is a one dimensional Brownian motion, $l_t$ the local time at 0 of $B_t$ and $Y_t=\left|B_t\right|+\ell_t$, a 3-dimensional Bessel process starting from 0 . There is the following beautiful result of Pitman:
$$E\left{f\left(\left|B_t\right|\right) \mid \sigma\left(Y_s: s \leq t\right)\right}=\int_0^1 f\left(x Y_t\right) \mathrm{d} x=V f\left(Y_t\right)$$
where $V$ is the Markov kernel: $V(x, \mathrm{~d} z)=\frac{\mathbf{1}_{0 \leq z \leq x}}{x} \mathrm{~d} z[2,21]$.
A second example, [11], which demonstrates the twist effect is on the product space of the circle. Let $p: S^1 \times S^1 \rightarrow S^1$ be the projection on the first factor. For $0<\alpha<\frac{\pi}{4}$, define the diffusion operator on $S^1 \times S^1$ :
$$\mathcal{B}=\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\tan \alpha \frac{\partial^2}{\partial x \partial y}$$
and the diffusion operator $\mathcal{A}=\frac{1}{2} \frac{\partial^2}{\partial x^2}$ on $S^1$. Then,
\begin{aligned} \mathcal{B}^V & =\frac{1}{2}\left(1-(\tan \alpha)^2\right) \frac{\partial^2}{\partial y^2} \ \mathcal{A}^H & =\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+(\tan \alpha)^2 \frac{\partial^2}{\partial y^2}\right)+\tan \alpha \frac{\partial^2}{\partial x \partial y} \end{aligned}

数学代写|随机分析代写Stochastic Calculus代考|Parallel Translation

The horizontal lift map $u_t$ can also be thought of solutions to:
$$\mathrm{d} u_t=\sum H\left(e_i\right)\left(u_t\right) \circ \mathrm{d} \sigma_t$$
In fact, if $\dot{v}t$ is the horizontal lift of $\dot{\sigma}_t, \dot{v}_t=\sum{i=1}^n\left\langle\dot{\sigma}t, e_i\right\rangle H\left(e_i\right)\left(\tilde{\sigma}_t\right)$. Note that, $/ / t(\sigma)$ is not a solution to a Markovian equation, the pair $\left(/ / t(\sigma), u_t\right)$ is. In local coordinates for $v_t^i$ the ith component of $/ / t(\sigma)(v), v \in T{\sigma_0} M$,
$$\mathrm{d} v_t^k=-\Gamma_{i, j}^k\left(\sigma_t\right) v_t^j \circ \mathrm{d} \sigma_t^i .$$
If $\sigma_t$ is the solution of the $\operatorname{SDE~} \mathrm{d} x_t^k=X_i^k\left(x_t\right) \circ \mathrm{d} B_t^i+X_0^k\left(x_t\right) \mathrm{d} t$, then
$$\mathrm{d} v_t^k=-\Gamma_{i, j}^k\left(x_t\right) v_t^j X_i^k\left(x_t\right) \circ \mathrm{d} B_t^i-\Gamma_{i, j}^k\left(x_t\right) v_t^j X_0^k\left(x_t\right) \mathrm{d} t$$

数学代写|随机分析代写Stochastic Calculus代考|The Diffeomorphism Group Example

$$h_f(v)(x)=\tilde{X}(f)\left(Y\left(f\left(x_0\right)\right) v\right)(x)=X(f(x))\left(Y\left(f\left(x_0\right)\right) v\right) .$$

$$E\left{f\left(\left|B_t\right|\right) \mid \sigma\left(Y_s: s \leq t\right)\right}=\int_0^1 f\left(x Y_t\right) \mathrm{d} x=V f\left(Y_t\right)$$

$$\mathcal{B}=\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\tan \alpha \frac{\partial^2}{\partial x \partial y}$$

\begin{aligned} \mathcal{B}^V & =\frac{1}{2}\left(1-(\tan \alpha)^2\right) \frac{\partial^2}{\partial y^2} \ \mathcal{A}^H & =\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+(\tan \alpha)^2 \frac{\partial^2}{\partial y^2}\right)+\tan \alpha \frac{\partial^2}{\partial x \partial y} \end{aligned}

数学代写|随机分析代写Stochastic Calculus代考|Parallel Translation

$$\mathrm{d} u_t=\sum H\left(e_i\right)\left(u_t\right) \circ \mathrm{d} \sigma_t$$

$$\mathrm{d} v_t^k=-\Gamma_{i, j}^k\left(x_t\right) v_t^j X_i^k\left(x_t\right) \circ \mathrm{d} B_t^i-\Gamma_{i, j}^k\left(x_t\right) v_t^j X_0^k\left(x_t\right) \mathrm{d} t$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 随机分析

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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数学代写|随机分析代写Stochastic Calculus代考|In Metric Form

Note that $\sigma^{\mathcal{A}}$ gives rise to a positive definite bilinear form on $T^* M$ :
$$\langle\phi, \psi\rangle_x=\phi(x)\left(\sigma_x^{\mathcal{A}}(\psi(x))\right)$$

and this induces an inner product on $E_x$ :
$$\langle u, v\rangle_x=\left(\sigma_x^{\mathcal{A}}\right)^{-1}(u)(v)$$
For an orthonormal basis $\left{e_i\right}$ of $E_x$, let $e_i^=\left(\sigma_x^{\mathcal{A}}\right)^{-1}\left(e_i\right)$. Then, $e_j^ \sigma^{\mathcal{A}}\left(e_i^\right)=$ $\left(\sigma_x^{\mathcal{A}}\right)^{-1}\left(e_j\right)\left(e_i\right)=\left\langle e_j, e_i\right\rangle$ and hence $$\langle\phi, \psi\rangle_x=\sum_i\left\langle e_j, e_i\right\rangle \phi\left(e_i\right) \psi\left(e_j\right)=\sum_i \phi\left(e_i\right) \psi\left(e_i\right) .$$ Likewise the symbol $\sigma^{\mathcal{A}^H}$ induces an inner product on $T^ N$ with the property that $\langle\phi \circ T p, \psi \circ T p\rangle=\langle\phi, \psi\rangle$ and a metric on $H \subset T N$ which is the same as that induced by $\mathrm{h}$ from $T M$. Note that $\sigma^{\mathcal{B}}=\sigma^{\mathcal{A}^H}+\sigma^{\mathcal{B}^V}$, where $\mathcal{B}^V$ is the vertical part of $\mathcal{B}$, and $\operatorname{Im}\left[\sigma^{\mathcal{B}^V}\right] \cap H={0}$. Let $\mu$ be an invariant measure for $\mathcal{A}^H$ and $\mu_M=p_*(\mu)$ the pushed forward measure which is an invariant measure for $\mathcal{A}$.
If $\mathcal{A}$ is symmetric,
\begin{aligned} \int_M\langle\mathrm{~d} f, \mathrm{~d} g\rangle \mu_M(\mathrm{~d} x) & =\int \sigma^{\mathcal{A}}(\mathrm{d} f, \mathrm{~d} g) \mu_M(\mathrm{~d} x) \ & =\frac{1}{2} \int[\mathcal{A}(f g)-f(\mathcal{A} g)-g(\mathcal{A} f)] \mu_M(\mathrm{~d} x) \ & =-\int_M f \mathcal{A} g \mathrm{~d} \mu_M(x) \end{aligned}

数学代写|随机分析代写Stochastic Calculus代考|On the Heisenberg Group

A Lie group is a group $G$ with a manifold structure such that the group multiplication $G \times G \rightarrow G$ and taking inverse are smooth. Its tangent space at the identity $g$ can be identified with left invariant vector fields on $G, X(a)=T L_a X(e)$ and we denote $A^*$ the left invariant vector field with value $A$ at the identity. The tangent space $T_a G$ at $a$ can be identified with $\mathrm{g}$ by the derivative $T L_a$ of the left translation map. Let $\alpha_t=\exp (t A)$ be the solution flow to the left invariant vector field $T L_a A$ whose value at 0 is the identity then it is also the flow for the corresponding right invariant vector field: $\dot{\alpha}s=\left.\frac{\mathrm{d}}{\mathrm{d} t}\right|{t=s} \exp ^{(t-s) A} \exp ^{s A}=T R_{\alpha_s} A$. Then $u_t=a \exp (t A)$ is the solution flow through $a$.

Consider the Heisenberg group $G$ whose elements are $(x, y, z) \in \mathbf{R}^3$ with group product
$$\left(x_1, y_1, z_1\right)\left(x_2, y_2, z_2\right)=\left(x_1+x_2, y_1+y_2, z_1+z_2+\frac{1}{2}\left(x_1 y_2-x_2 y_1\right)\right) .$$
The Lie bracket operation is $\left[(a, b, c),\left(a^{\prime}, b^{\prime}, c^{\prime}\right)\right]=\left(0,0, a b^{\prime}-a^{\prime} b\right)$. Note that for $X, Y \in \mathrm{g}, \mathrm{e}^X \mathrm{e}^Y=\mathrm{e}^{X+Y+\frac{1}{2}[X, Y]}$. If $A=(a, b, c)$, then $A^*=\left(a, b, c+\frac{1}{2}(x b-\right.$ $y a)$ ). Consider the projection $\pi: G \rightarrow \mathbf{R}^2$ where $\pi(x, y, z)=(x, y)$. Let
\begin{aligned} & X_1(x, y, z)=\left(1,0,-\frac{1}{2} y\right), \quad X_2(x, y, z)=\left(0,1, \frac{1}{2} x\right) \ & X_3(x, y, z)=(0,0,-1) \end{aligned}
be the left invariant vector fields corresponding to the standard basis of $\mathrm{g}$. The vector spaces $H_{(x, y, z)}=\operatorname{span}\left{X_1, X_2\right}=\left{\left(a, b, \frac{1}{2}(x b-y a)\right)\right}$ are of rank 2. They are the horizontal tangent spaces associated to the Laplacian $\mathcal{A}=\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)$ on $\mathbf{R}^2$ and the left invariant Laplacian $\mathcal{B}:=\frac{1}{2} \sum_{i=1}^3 L_{X_i} L_{X_i}$ on $G$. The vertical tangent space is ${(0,0, c)}$, and there is a horizontal lifting map from $T_{(x, y)} \mathbf{R}^2$ :
$$h_{(x, y, z)}(a, b)=\left(a, b, \frac{1}{2}(x b-y a)\right)$$

数学代写|随机分析代写Stochastic Calculus代考|In Metric Form

$$\langle\phi, \psi\rangle_x=\phi(x)\left(\sigma_x^{\mathcal{A}}(\psi(x))\right)$$

$$\langle u, v\rangle_x=\left(\sigma_x^{\mathcal{A}}\right)^{-1}(u)(v)$$

\begin{aligned} \int_M\langle\mathrm{~d} f, \mathrm{~d} g\rangle \mu_M(\mathrm{~d} x) & =\int \sigma^{\mathcal{A}}(\mathrm{d} f, \mathrm{~d} g) \mu_M(\mathrm{~d} x) \ & =\frac{1}{2} \int[\mathcal{A}(f g)-f(\mathcal{A} g)-g(\mathcal{A} f)] \mu_M(\mathrm{~d} x) \ & =-\int_M f \mathcal{A} g \mathrm{~d} \mu_M(x) \end{aligned}

数学代写|随机分析代写Stochastic Calculus代考|On the Heisenberg Group

$$\left(x_1, y_1, z_1\right)\left(x_2, y_2, z_2\right)=\left(x_1+x_2, y_1+y_2, z_1+z_2+\frac{1}{2}\left(x_1 y_2-x_2 y_1\right)\right) .$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。