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# 数学代写|随机分析代写Stochastic Calculus代考|Diffusions on the Euclidean Space

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## 数学代写|随机分析代写Stochastic Calculus代考|Diffusions on the Euclidean Space

Take the example that $N=\mathbf{R}^2$ and $M=\mathbf{R}$. Any elliptic diffusion operators on $M$ is of the form $a(x) \frac{\mathrm{d}^2}{\mathrm{dx}}$ and a diffusion operator on $N$ is of the form $\mathcal{B}=$ $a(x, y) \frac{\mathrm{d}^2}{\mathrm{~d} x^2}+d(x, y) \frac{\mathrm{d}^2}{\mathrm{~d} y^2}+c(x, y) \frac{\mathrm{d}^2}{\mathrm{~d} x \mathrm{~d} y}$ with $4 a d>c^2$ and $a>0$. Now, $\mathcal{B}$ is over $\mathcal{A}$ implies that $a(x, y)=a(x)$ for all $y$. If $a, b, c$ are constants, a change of variable of the form $x=u$ and $y=(c / 2 \sqrt{a}) u+v$ transforms $\mathcal{B}$ to $a^2 \frac{\partial^2}{\partial u^2}+\left(d-c^2 / 4 a\right) \frac{\partial^2}{\partial v^2}$. In this local coordinates, $\mathcal{B}$ and $\mathcal{A}$ have a trivial projective relation. In general, we may seek a diffeomorphism $\Phi:(x, y) \mapsto(u, v)$ so that $\Phi$ intertwines $\mathcal{B}$ and $\tilde{B}$ where $\tilde{B}$ is the sum of $a^2 \frac{\partial^2}{\partial u^2}$ and an operator of the form $\frac{\partial^2}{\partial v^2}$. This calculation is quite messy. However, according to the theory in [11], the horizontal lifting map
$$v \mapsto \sigma^{\mathcal{B}}(T p)^\left(\sigma^{\mathcal{A}}\right)^{-1}(v)=\sigma^{\mathcal{B}}\left(\frac{v}{a}, 0\right)^T=\left(\begin{array}{cc} a & \frac{c}{2} \ \frac{c}{2} & d \end{array}\right)\left(\begin{array}{c} \frac{v}{a} \ 0 \end{array}\right)=\left(v, \frac{c}{2 a} v\right) .$$ where $p:(x, y) \rightarrow x$ and $T p$ is the derivative map and $(T p)^$ is the corresponding adjoint map. Hence, the lifting of $\mathcal{A}$, as the square of the lifting $\sqrt{a} \frac{\mathrm{d}}{\mathrm{d} x}$ gives $\sqrt{a}\left(\frac{\mathrm{d}}{\mathrm{d} x}+\frac{c}{2 a} \frac{\mathrm{d}}{\mathrm{d} y}\right)$ and resulting the completion of the square procedure and the splitting of $\mathcal{B}$ :
$$\mathcal{B}=a\left(\frac{\mathrm{d}}{\mathrm{d} x}+\frac{c}{2 a} \frac{\mathrm{d}}{\mathrm{d} y}\right)^2+\left(d-\frac{c^2}{4 a}\right) \frac{\mathrm{d}^2}{\mathrm{~d} y^2} .$$

## 数学代写|随机分析代写Stochastic Calculus代考|The SDE Example and the Associated Connection

Consider SDE (1). For each $y \in M$, define the linear map $X(y)(e): \mathbf{R}^m \rightarrow T_y M$ by $X(y)(e)=\sum_{i=1}^m X_i(y)\left\langle e, e_i\right\rangle$. Let $Y(y): T_y M \rightarrow[\operatorname{ker} X(y)]^{\perp}$ be the right inverse to $X(y)$. The symbol of the generator $\mathcal{A}$ is $\sigma_y^{\mathcal{A}}=\frac{1}{2} X(y) X(y)^*$, which induces a Riemannian metric on the manifold in the elliptic case, and a sub-Riemannian metric in the case of $\sigma^{\mathcal{A}}$ being of constant rank .

This map $X$ also induces an affine connection $\breve{\nabla}$, which we called the LW connection, on the tangent bundle which is compatible with the Riemannian metric it induced as below. If $v \in T_{y_0} M$ is a tangent vector and $U \in \Gamma T M$ a vector field,
$$\left(\breve{\nabla}_v U\right)\left(y_0\right)=X\left(y_0\right) D(Y(y) U(y))(v)$$
At each point $y \in M$, the linear map
$$X(y): \mathbf{R}^m=\operatorname{ker} X(y) \oplus[\operatorname{ker} X(y)]^{\perp} \rightarrow T_y M$$

induces a direct sum decomposition of $\mathbf{R}^m$. The connection defined above is a metric connection with the property that
$$\breve{\nabla}v X(e) \equiv 0, \quad \forall e \in\left[\operatorname{ker} X\left(y_0\right)\right]^{\perp}, v \in T{y_0} M .$$
This connection is the adjoint connection by the induced diffusion pair on the general linear frame bundle mentioned earlier. See [9] where it is stated any metric connection on $M$ can be defined through an SDE, using Narasimhan and Ramanan’s universal connection.

## MATLAB代写

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