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姓名 陳韋達(Wei-Da Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱
(On Space-Time Harmonic Functions for Gaussian Diffusion Processes)
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摘要(中) 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
[
egin{cases}
dX_{t}=BX_{t}dt+dW_{t},\
X_{0}=x_{0},
end{cases}
]
其中$B$是個$d imes d$矩陣, $W$是個$d$維布朗運動,而$x_{0}inmathbb{R}^{d}$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
egin{align*}
(frac{partial}{partial t}+frac{1}{2} riangle+Bxcdot abla)g=0,mbox{ }g>0mbox{ on }(0,infty) imesmathbb{R}^{d}.
end{align*}
$g$的積分表現式是
egin{align*}
g(t,x) & =int_{mathbb{R}^{d}}M_{B}(t,x;z)
ho(dz),
end{align*}
其中$
ho$是個機率分布且${M_{B}(cdot,cdot;z);zinmathbb{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}$滿足
[
egin{cases}
dX_{t}^{g}=frac{ abla g(t,X_{t}^{g})}{g(t,X_{t}^{g})}dt+Bcdot X_{t}^{g}dt+dW_{t},\
X_{0}^{g}=x_{0}.
end{cases}
]
我們研究$X_{t}^{g}$的極限行為當$t
ightarrowinfty$。我們先得到一個有趣的$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
ightarrowinfty$。也就是說
egin{align*}
frac{X_{t}^{g}}{t}
ightarrow Y,mbox{ as }t
ightarrowinfty,
end{align*}
其中$Y$是個隨機變數。此外,$X_{t}^{g}$有令人意外的表現式
egin{align*}
X_{t}^{g} & =x_{0}+tY+widehat{W}_{t},
end{align*}
其中$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$的積分表現式,其中$
ho$(在表現式裡)是$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。
摘要(英) 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
[
egin{cases}
dY_{t}=BY_{t}dt+dW_{t},\
Y_{0}=x_{0},
end{cases}
]
where $B$ is a $d imes d$ matrix, $W$ is a $d-$dimensional Brownian
motion, and $x_{0}inmathbb{R}^{d}$ is the initial value of $Y$.
$g$ is a positive space-time harmonic function for $Y_{t}$ which
satisfies
egin{align*}
(frac{partial}{partial t}+frac{1}{2} riangle+Bxcdot abla)g=0,mbox{ }g>0mbox{ on }(0,infty) imesmathbb{R}^{d}.
end{align*}
The integral formula of $g$ is given by
egin{align*}
g(t,x) & =int_{mathbb{R}^{d}}M_{B}(t,x;z)
ho(dz),
end{align*}
where $
ho$ is a probability distribution and ${M_{B}(cdot,cdot;z);zinmathbb{R}^{d}}$
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
[
egin{cases}
dX_{t}=frac{ abla g(t,X_{t})}{g(t,X_{t})}dt+Bcdot X_{t}dt+dW_{t},\
X_{0}=x_{0}.
end{cases}
]
We study the limiting behavior of $X_{t}$ as $t
ightarrowinfty$.
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 $t
ightarrowinfty$. This means
egin{align*}
frac{X_{t}^{g}}{t}
ightarrow Y,mbox{ as }t
ightarrowinfty,
end{align*}
where $Y$ is a random variable. Futhermore, $X_{t}$ has remarkable
representation
egin{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 $
ho$
(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.
關鍵字(中) ★ 時間空間調和函數
★ 高斯擴散過程
關鍵字(英)
論文目次 Contents
Abstract(Chinese) ii
Abstract iii
Part 1 1
1 Introduction 1
2 The problem and main results 3
3 Proof of Theorem 12
3.1 Proof of Theorem 1 13
3.2 Proof of Theorem 2 16
4 Applications 20
4.1 Find harmonic function by representation 20
4.2 Boundary-Crossing probability 21
References of Part 1 30
Part 2 32
1 Introduction 32
2 The problem and main results 34
3 Proof of Theorems 43
3.1 Proof of Theorem 1 43
3.2 Proof of Theorem 2 53
4 Applications 56
4.1 Representation of harmonic function 56
4.1.1 Case1: B = 0 57
4.1.2 Case2: The real part of all eigenvalues of B are positive 58
4.1.3 Case3: The real part of all eigenvalues of B are zero 64
4.1.4 Case4: The real part of all eigenvalues of B are negative 66
4.2 Boundary-Crossing probability 68
5 Appendix 76
5.1 Reference for Riccati Equation 77
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指導教授 許順吉 審核日期 2017-8-24
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