English  |  正體中文  |  简体中文  |  全文筆數/總筆數 : 62278/62278 (100%)
造訪人次 : 17925778      線上人數 : 130
RC Version 7.0 © Powered By DSPACE, MIT. Enhanced by NTU Library IR team.
搜尋範圍 查詢小技巧:
  • 您可在西文檢索詞彙前後加上"雙引號",以獲取較精準的檢索結果
  • 若欲以作者姓名搜尋,建議至進階搜尋限定作者欄位,可獲得較完整資料
  • 進階搜尋


    請使用永久網址來引用或連結此文件: http://ir.lib.ncu.edu.tw/handle/987654321/74976


    題名: On Space-Time Harmonic Functions for Gaussian Diffusion Processes
    作者: 陳韋達;Chen, Wei-Da
    貢獻者: 數學系
    關鍵詞: 時間空間調和函數;高斯擴散過程
    日期: 2017-08-24
    上傳時間: 2017-10-27 16:14:33 (UTC+8)
    出版者: 國立中央大學
    摘要: 在本論文中,我們研讀對應高斯擴散過程的正值時間空間調合函數之積分表現式。高斯擴散過程$X_{t}$在$\mathbb{R}^{d}$上面滿足
    \[
    \begin{cases}
    dX_{t}=BX_{t}dt+dW_{t},\\
    X_{0}=x_{0},
    \end{cases}
    \]
    其中$B$是個$d\times d$矩陣, $W$是個$d$維布朗運動,而$x_{0}\in\mathbb{R}^{d}$是$X$的初始值。$g$是個正值時間空間調合函數對應到隨機過程$X_{t}$且滿足
    \begin{align*}
    (\frac{\partial}{\partial t}+\frac{1}{2}\triangle+Bx\cdot\nabla)g=0,\mbox{ }g>0\mbox{ on }(0,\infty)\times\mathbb{R}^{d}.
    \end{align*}
    $g$的積分表現式是
    \begin{align*}
    g(t,x) & =\int_{\mathbb{R}^{d}}M_{B}(t,x;z)\rho(dz),
    \end{align*}
    其中$\rho$是個機率分布且$\{M_{B}(\cdot,\cdot;z);z\in\mathbb{R}^{d}\}$是一系列獨立於$g$的函數。為了獲得表現式,我們考慮一個跟$g$有關的隨機過程$X_{t}^{g}$,其中$X_{t}^{g}$滿足
    \[
    \begin{cases}
    dX_{t}^{g}=\frac{\nabla g(t,X_{t}^{g})}{g(t,X_{t}^{g})}dt+B\cdot X_{t}^{g}dt+dW_{t},\\
    X_{0}^{g}=x_{0}.
    \end{cases}
    \]
    我們研究$X_{t}^{g}$的極限行為當$t\rightarrow\infty$。我們先得到一個有趣的$X_{t}^{g}$表現式。然後$g$的積分表現式自然可以從$X_{t}^{g}$表現式獲得。在第一部分,我們考慮布朗運動,也就是$B=0$。在這個案例裡,我們證明$X_{t}^{g}$有線性成長速度$Y$當$t\rightarrow\infty$。也就是說
    \begin{align*}
    \frac{X_{t}^{g}}{t}\rightarrow Y,\mbox{ as }t\rightarrow\infty,
    \end{align*}
    其中$Y$是個隨機變數。此外,$X_{t}^{g}$有令人意外的表現式
    \begin{align*}
    X_{t}^{g} & =x_{0}+tY+\widehat{W}_{t},
    \end{align*}
    其中$\widehat{W}_{t}$是個獨立於$Y$的布朗運動。利用這個結果,我們獲得$g$的積分表現式,其中$\rho$(在表現式裡)是$Y$的機率分布。在第二部分,我們考慮一般的$B$。我們利用類似的方法去獲得$X_{t}^{g}$的不同成長速度和$X_{t}^{g}$表現式。然後我們可以得到$g$的積分表現式。我們也討論一些時間空間調合函數的積分表現式的應用。第一個例子是看正值(空間)調合函數的積分表現式。第二個例子是用來看邊界穿越機率的計算。;In this dissertation, we study the integral representation of positive
    space-time harmonic function for Gaussian diffusion processes. A Gaussian
    diffusion process $Y_{t}$ in $\mathbb{R}^{d}$ is governed by
    \[
    \begin{cases}
    dY_{t}=BY_{t}dt+dW_{t},\\
    Y_{0}=x_{0},
    \end{cases}
    \]
    where $B$ is a $d\times d$ matrix, $W$ is a $d-$dimensional Brownian
    motion, and $x_{0}\in\mathbb{R}^{d}$ is the initial value of $Y$.
    $g$ is a positive space-time harmonic function for $Y_{t}$ which
    satisfies
    \begin{align*}
    (\frac{\partial}{\partial t}+\frac{1}{2}\triangle+Bx\cdot\nabla)g=0,\mbox{ }g>0\mbox{ on }(0,\infty)\times\mathbb{R}^{d}.
    \end{align*}
    The integral formula of $g$ is given by
    \begin{align*}
    g(t,x) & =\int_{\mathbb{R}^{d}}M_{B}(t,x;z)\rho(dz),
    \end{align*}
    where $\rho$ is a probability distribution and $\{M_{B}(\cdot,\cdot;z);z\in\mathbb{R}^{d}\}$
    is a family of functions which is independent of $g$. To obtain such
    integral representation, we consider a process associated to $g$
    deduced by $X_{t}$ which is governed by
    \[
    \begin{cases}
    dX_{t}=\frac{\nabla g(t,X_{t})}{g(t,X_{t})}dt+B\cdot X_{t}dt+dW_{t},\\
    X_{0}=x_{0}.
    \end{cases}
    \]
    We study the limiting behavior of $X_{t}$ as $t\rightarrow\infty$.
    We first obtain an interesting representation of $X_{t}$. Then the
    integral formula of $g$ will follow. In Part 1, we consider the Brownian
    motion, where $B=0$. In this case, we show $X_{t}$ has linear growth
    with the rate given by $Y$ as $t\rightarrow\infty$. This means
    \begin{align*}
    \frac{X_{t}^{g}}{t}\rightarrow Y,\mbox{ as }t\rightarrow\infty,
    \end{align*}
    where $Y$ is a random variable. Futhermore, $X_{t}$ has remarkable
    representation
    \begin{align*}
    X_{t} & =x_{0}+tY+\widehat{W}_{t},
    \end{align*}
    where $\widehat{W}_{t}$ is a Brownian motion independent of $Y$.
    Using this, we obtain an integral representation for $g$, where $\rho$
    (in the representation) is the disrtibution of $Y$. In Part 2, we
    consider general $B$. We apply the similar approach to obtain the
    growth of $X_{t}$, with different rate and a representation of $X_{t}$.
    Then we can obtain the integral representation formula of $g$. We
    also discuss some applications of the integral representation of space-time
    harmonic functions. The first example is the integral representation
    for a positive (space) harmonic functions. The second example is the
    use in the calculation of the boundary crossing probability.
    顯示於類別:[數學研究所] 博碩士論文

    文件中的檔案:

    檔案 描述 大小格式瀏覽次數
    index.html0KbHTML138檢視/開啟


    在NCUIR中所有的資料項目都受到原著作權保護.

    社群 sharing

    ::: Copyright National Central University. | 國立中央大學圖書館版權所有 | 收藏本站 | 設為首頁 | 最佳瀏覽畫面: 1024*768 | 建站日期:8-24-2009 :::
    DSpace Software Copyright © 2002-2004  MIT &  Hewlett-Packard  /   Enhanced by   NTU Library IR team Copyright ©   - 回饋  - 隱私權政策聲明