### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT4061

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Range Inclusion and the Duality Argument

Clearly, any controllability problem (formulated in Definition $1.1$ or more generally, in Definition $5.6$ in Chapter 5 ) can be viewed as an equation problem, in which both the state $x(\cdot)$ and the control $u(\cdot)$ variables are unknowns. Namely, instead of viewing $u(\cdot)$ as a control variable, we may simply regard it as another unknown variable 2 . One of the main concerns in this book is to study the controllability problems for linear stochastic evolution equations. As we shall see later, this is far from an easy task.

It is easy to see that, in many cases solving linear equations are equivalent to showing range inclusion for suitable linear operators. Because of this, we shall present below two known range inclusion theorems (i.e., Theorems $1.7$ and $1.10$ below) in an abstract setting.

Throughout this section, $X, Y$ and $Z$ are Banach spaces. Denote by $\mathcal{L}(X ; Y)$ the Banach space of all bounded linear operators from $X$ to $Y$, with the usual operator norm. When $X=Y$, we simply write $\mathcal{L}(X)$ instead of $\mathcal{L}(X ; X)$. For any $L \in \mathcal{L}(X ; Y)$, denote by $\mathcal{R}(L)$ the range of $L$. We begin with the following result.

Theorem 1.7. Let $F \in \mathcal{L}(X ; Z)$ and $G \in \mathcal{L}(Y ; Z)$. The following assertions hold:
1) If $\mathcal{R}(F) \supseteq \mathcal{R}(G)$, then there is a constant $\mathcal{C}>0$ such that
$$\left|G^{} z^{}\right|{Y^{}} \leq \mathcal{C}\left|F^{} z^{}\right|{X^{}}, \quad \forall z^{} \in Z^{} .$$
2) If $X$ is reflexive and (1.13) holds for some constant $\mathcal{C}>0$, then $\mathcal{R}(F) \supseteq$ $\mathcal{R}(G)$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Two Basic Methods in This Book

In this section, we shall present two basic methods (via illuminating examples) that will be systematically used throughout this book.

The main method that we employ in this book to deal with the analysis of the structure of stochastic distributed parameter systems is the global Carleman type estimate. This method was introduced by T. Carleman ([47]) in 1939 to prove the uniqueness of solutions to second order elliptic partial differential equations with two variables. The key in [47] is an elementary energy estimate with some exponential weight. This type of weighted energy estimates, now referred to as Carleman estimates, have become one of the major tools in the study of unique continuation property, inverse problems and control problems for many partial differential equations. However, it is only in the last ten plus years that the power of the global Carleman estimate in the context of controllability of stochastic partial differential equations came to be realized. For the readers’ convenience, we explain the main idea of Carleman estimate by the following very simple example:

Example 1.14. Consider the following ordinary differential equation in $\mathbb{R}^{n}$ :
$$\left{\begin{array}{l} y_{t}(t)=a(t) y(t) \quad \text { in }[0, T], \ y(0)=y_{0} . \end{array}\right.$$
It is well-known that if $a \in L^{\infty}(0, T)$, then there is a constant $\mathcal{C}_{T}>0$ such that for all solutions of (1.45), it holds that $$\max {t \in[0, T]}|y(t)|{\mathbb{R}^{n}} \leq \mathcal{C}{T}\left|y{0}\right|{\mathbb{R}^{n}}, \quad \forall y{0} \in \mathbb{R}^{n} .$$
Now we give a slightly different proof of this result via Carleman-type estimate:
For any $\lambda \in \mathbb{R}$, it is easy to see that
\begin{aligned} &\frac{d}{d t}\left(e^{-2 \lambda t}|y(t)|{\mathbb{R}^{n}}^{2}\right) \ &=-2 \lambda e^{-2 \lambda t}|y(t)|{\mathbb{R}^{n}}^{2}+2 e^{-2 \lambda t}\left\langle y_{t}(t), y(t)\right\rangle_{\mathbb{R}^{n}}=2(a(t)-\lambda) e^{-2 \lambda t}|y(t)|{\mathbb{R}^{n}}^{2} . \end{aligned} Choosing $\lambda=|a|{L^{\infty}(0, T)}$, we find that
$$|y(t)|{\mathbb{R}^{n}} \leq e^{\lambda T}\left|y{0}\right|_{\mathbb{R}^{n}}, \quad t \in[0, T],$$
which proves (1.46).

## 数学代写|随机过程统计代写Stochastic process statistics代考|Range Inclusion and the Duality Argument

1) 如果 $\mathcal{R}(F) \supseteq \mathcal{R}(G)$, 那么有一个常数 $\mathcal{C}>0$ 这样
$$|G z| Y \leq \mathcal{C}|F z| X, \quad \forall z \in Z$$
2) 如果 $X$ 是自反的并且 (1.13) 对某个常数成立 $\mathcal{C}>0$ ，然后 $\mathcal{R}(F) \supseteq \mathcal{R}(G)$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Two Basic Methods in This Book

$\$ \$$左 {$$
y_{t}(t)=a(t) y(t) \quad \text { in }[0, T], y(0)=y_{0} .
$$【正确的。 Itiswell – knownthatif \ a \in L^{\infty}(0, T) \$$, thenthereisaconstant $\$ \mathcal{C}{T}>0$\$suchthat forallsolutionsof $\operatorname{Imax}{t \backslash \operatorname{lin}[0, T]}|\mathrm{y}(\mathrm{t})|{\backslash \operatorname{mathbb}{R} \wedge{n}} \backslash \operatorname{leq} \backslash$ mathcal ${C}{T} \backslash \operatorname{left} \mid \mathrm{y}{0} \backslash$ right $\mid{\backslash$ mathbb ${R} \wedge{n}}$, Iquad $\backslash$ forall $y{0} \backslash$ in $\backslash m a t h b b{R} \wedge{n}$ 。 NowwegiveaslightlydifferentproofofthisresultviaCarleman-typeestimate: Forany $\$ \lambda \in \mathbb{R} \$$, it$$ \frac{d}{d t}\left(e^{-2 \lambda t}|y(t)| \mathbb{R}^{n 2}\right) \quad=-2 \lambda e^{-2 \lambda t}|y(t)| \mathbb{R}^{n 2}+2 e^{-2 \lambda t}\left\langle y{t}(t), y(t)\right\rangle_{\mathbb{R}^{n}}=2(a(t)-\lambda) e^{-2 \lambda t}|y(t)| \mathbb{R}^{n 2}
$$Choosing \ \lambda=|a| L^{\infty}(0, T) \$$, we findthat $\$ \

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