关于非退化扩散过程的几点注记
|
摘要
关于非退化扩散过程的研究，已有一些结果。文[1]得到了N维(N≥2)非退化扩散过程样本轨道的象集的Hausdorff维数，而对N=1时，只得到了其象集的Hausdorff维数的一个上界估计(见[2])。本文对N≥1时，得到了象集的Hausdorff维数，但证明方法则有别于文[1]。这是对非退化扩散过程所做的第一点注记。
文[3]证明了对于一维Brown运动，几乎所有的样本轨道的水平集的Hausdorff维数是1/2。本文获得了一维非退化扩散过程的样本轨道的水平集的Hausdorff维数，其结果类似于Brown运动。进一步，研究了非退化扩散过程的样本轨道的逆像集的Hausdorff维数。这是对非退化扩散过程所做的第二点注记。
众所周知，对N维Brown运动，当N≥4时，几乎所有的样本轨道不存在二重点，而当N≤3时，几乎所有的样本轨道存在二重点。对定义在完备概率申间(Ω，F，P)上取值于R~N的Gauss随机场{X(t，ω)：t∈R~D，ω∈Ω}，其分量相互独立且具有E[(X_i(t)-X_i(s))~2]=|t-s|~(2α)0＜α＜1，1≤i≤N的性质，当αN＞2D时，几乎所有的样本轨道不存在二重点，而当αN＜2D时，几乎所有的样本轨道存在二重点(见[4])。从这得到启示，我们开始考虑N维非退化扩散过程的样本轨道是否具有同样的性质。事实上，我们讨论了其二重点问题。这是对非退化扩散过程所做的第三点注记。
具体说来，得到如下主要结果：
1．设α(x)和β(x)满足条件C，X(t)是方程(1-1)的解，E为有界闭集，且E R_+。则：
d ioX(E)=min{N，ZdimE}a.s，
2.设N二1，。(x)不flp(x)满足条件C，X(t)足jJ‘程(1一l)的解，E为有界
f刁J集，且E二R+.则:
dimX(E)=min{l，ZdimE}a.5.
3.设N=1，a(x)和p(x)满足条件C，X(t)是方程(l一l)的解，E为有界
闭集，E二R+且dimE扒/2二则:
d imX(E)=1 a.s
注:此结论3表明当dimE全1/2时，一维非退化扩散过程的象集的
Hausdorff维数达到了址大值。
遭.设N二l，。(x)和p(x)满足条件C，X(t)是方程(l一l)的解。则:
d imX一，(x)=粤a.5.(丫xoR)
5.设N二l，a(x)和p(x)满足条件C，X(t)是方程(1一1)的解。B为R’中
、、.刀产口
B护刀
不沐B
/‘、，
Borel集，令X
d 1 mX
={t)O:X(t)eB}。则:
一l+dim(B)
2
6.设。(x)不「Ip(x)满足条件C，X(t)是方程(l一1)的解。则:
当N之5 11寸，有:
L:二沪
其，}，L，二{x任R、:日不同的t
a .5.
t:任R，，使得x(t.，。)=x(tZ
。)=x}
There are some results' on the study of a N-dimensional Nondegenerate Diffusion Process. In [1], for N ≥ 2, the Hausdorff dimension of image set of its sample path is obtained, but for N=l, only its upper bound estimate (See [2]). In this paper, we obtained the Hausdorff dimension of image set for N≥1. However, the proving method is distinct from [1]. This is the first note on the Nondegenerate Diffusion Process.
In [3], it was shown that the Hausdorff dimension of level set of one-dimensional Brownian Sample Paths is 1/2 with probability 1. This paper obtained the result on the Hausdorff dimension of level set of one-dimensional Nondegenerate Diffusion Process, which resembles that of the Brownian motion. Futhermore, we studied that the Hausdorff dimension of its inverse image set. This is the second note on the Nondegenerate Diffusion Process.
It is well known that, the Brownian paths in N-space have no double points with probability 1 if N≥4, butforN≤3, there are double points with probability 1. Let {X(t, w): t∈RD, w∈Ω} be a path continuous centered Gaussian random field defined on a complete probability space (Ω,F, P) and take values in RN such that E[(X,. (t)-X, (s))2 ] = | t-s|2a 02D, then almost all sample paths {X(t, w): t∈RD, (w∈Ω} have no double points of any length. But, if aN<2D, there are
double points with probability 1 (See [4]). From this, we begin to consider whether the N-dimensional Nondegenerate Diffusion Process possesses the same property. In fact, we discussed its double points problem. This is the third note on the Nondegenerate Diffusion Process.
Concretely speaking, the main results are obtained as follows:
1. Let a (x) and 13 (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. then,
dimX(E)=min{ N, 2dimE} a. s.
2. Let N=1, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rf. then,
dimX(E)=min{ 1, 2dimE} a. s.
3. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. dimE>1/2. then, dimX(E)= 1 a. s.
Remark. The result 3 shows that when dimE>1/2, the Hausdorff dimension of image set of one-dimensional Nondegenerate Diffusion Process takes the maximum value.
4. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). then, dimX-1(x)=1/2 a. s. (x∈R)
5. Let N=1, α (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let B be a borel set in R1. Let X-1 (B) = {t
≥0: X(t)∈B}. then, dimX-1 (B) = 1 + dim(B)/2 a. s.
?
6. Let a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). If N≥5, then, L2=φ a. s. where L , ={x ∈ R N ; 3 distinct t , , t , ∈ R + , such that
X(t1, w)=X( t2,w)=x}
文[3]证明了对于一维Brown运动，几乎所有的样本轨道的水平集的Hausdorff维数是1/2。本文获得了一维非退化扩散过程的样本轨道的水平集的Hausdorff维数，其结果类似于Brown运动。进一步，研究了非退化扩散过程的样本轨道的逆像集的Hausdorff维数。这是对非退化扩散过程所做的第二点注记。
众所周知，对N维Brown运动，当N≥4时，几乎所有的样本轨道不存在二重点，而当N≤3时，几乎所有的样本轨道存在二重点。对定义在完备概率申间(Ω，F，P)上取值于R~N的Gauss随机场{X(t，ω)：t∈R~D，ω∈Ω}，其分量相互独立且具有E[(X_i(t)-X_i(s))~2]=|t-s|~(2α)0＜α＜1，1≤i≤N的性质，当αN＞2D时，几乎所有的样本轨道不存在二重点，而当αN＜2D时，几乎所有的样本轨道存在二重点(见[4])。从这得到启示，我们开始考虑N维非退化扩散过程的样本轨道是否具有同样的性质。事实上，我们讨论了其二重点问题。这是对非退化扩散过程所做的第三点注记。
具体说来，得到如下主要结果：
1．设α(x)和β(x)满足条件C，X(t)是方程(1-1)的解，E为有界闭集，且E R_+。则：
d ioX(E)=min{N，ZdimE}a.s，
2.设N二1，。(x)不flp(x)满足条件C，X(t)足jJ‘程(1一l)的解，E为有界
f刁J集，且E二R+.则:
dimX(E)=min{l，ZdimE}a.5.
3.设N=1，a(x)和p(x)满足条件C，X(t)是方程(l一l)的解，E为有界
闭集，E二R+且dimE扒/2二则:
d imX(E)=1 a.s
注:此结论3表明当dimE全1/2时，一维非退化扩散过程的象集的
Hausdorff维数达到了址大值。
遭.设N二l，。(x)和p(x)满足条件C，X(t)是方程(l一l)的解。则:
d imX一，(x)=粤a.5.(丫xoR)
5.设N二l，a(x)和p(x)满足条件C，X(t)是方程(1一1)的解。B为R’中
、、.刀产口
B护刀
不沐B
/‘、，
Borel集，令X
d 1 mX
={t)O:X(t)eB}。则:
一l+dim(B)
2
6.设。(x)不「Ip(x)满足条件C，X(t)是方程(l一1)的解。则:
当N之5 11寸，有:
L:二沪
其，}，L，二{x任R、:日不同的t
a .5.
t:任R，，使得x(t.，。)=x(tZ
。)=x}
There are some results' on the study of a N-dimensional Nondegenerate Diffusion Process. In [1], for N ≥ 2, the Hausdorff dimension of image set of its sample path is obtained, but for N=l, only its upper bound estimate (See [2]). In this paper, we obtained the Hausdorff dimension of image set for N≥1. However, the proving method is distinct from [1]. This is the first note on the Nondegenerate Diffusion Process.
In [3], it was shown that the Hausdorff dimension of level set of one-dimensional Brownian Sample Paths is 1/2 with probability 1. This paper obtained the result on the Hausdorff dimension of level set of one-dimensional Nondegenerate Diffusion Process, which resembles that of the Brownian motion. Futhermore, we studied that the Hausdorff dimension of its inverse image set. This is the second note on the Nondegenerate Diffusion Process.
It is well known that, the Brownian paths in N-space have no double points with probability 1 if N≥4, butforN≤3, there are double points with probability 1. Let {X(t, w): t∈RD, w∈Ω} be a path continuous centered Gaussian random field defined on a complete probability space (Ω,F, P) and take values in RN such that E[(X,. (t)-X, (s))2 ] = | t-s|2a 0
double points with probability 1 (See [4]). From this, we begin to consider whether the N-dimensional Nondegenerate Diffusion Process possesses the same property. In fact, we discussed its double points problem. This is the third note on the Nondegenerate Diffusion Process.
Concretely speaking, the main results are obtained as follows:
1. Let a (x) and 13 (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. then,
dimX(E)=min{ N, 2dimE} a. s.
2. Let N=1, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rf. then,
dimX(E)=min{ 1, 2dimE} a. s.
3. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. dimE>1/2. then, dimX(E)= 1 a. s.
Remark. The result 3 shows that when dimE>1/2, the Hausdorff dimension of image set of one-dimensional Nondegenerate Diffusion Process takes the maximum value.
4. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). then, dimX-1(x)=1/2 a. s. (x∈R)
5. Let N=1, α (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let B be a borel set in R1. Let X-1 (B) = {t
≥0: X(t)∈B}. then, dimX-1 (B) = 1 + dim(B)/2 a. s.
?
6. Let a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). If N≥5, then, L2=φ a. s. where L , ={x ∈ R N ; 3 distinct t , , t , ∈ R + , such that
X(t1, w)=X( t2,w)=x}
引文
[1] 杨新建．多维非退化扩散过程样本的象集的Hausdorff维数．湖南师范大学学报．1994．17(2)：20-22
[2] 杨新建．扩散过程样本的Hlder连续性及其应用．湖南师范大学学报．1995．18(2)：13-18
[3] 胡迪鹤等．随机分形引论．武汉，武汉大学出版社．1995．100
[4] Norio Kono. Double points of a Gaussian Sample Path. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 45 (1978): 175-180
[5] LkedaN, WatanabeS. Stochastic Differential Equations and Diffusion Processes. North-Holland Publishing Company New York. 1981
[6] Sheu SJ. Some Estimate of the Transition Density of a Nondegenerate Diffusion Markov Process. Ann, Probab. 1991(2): 538-561
[7] Falconer K. Fractal Geometry, England, 1990
[8] 骆顺龙．张正敏．一类半鞅样本轨道的Fractal性质．应用数学 1996，9(1)：42-45
[9] D. W. Stroock An Introduction to the Theory of Large Deviations. Springer. Verlag, New York, 1984, 81-82.
[2] 杨新建．扩散过程样本的Hlder连续性及其应用．湖南师范大学学报．1995．18(2)：13-18
[3] 胡迪鹤等．随机分形引论．武汉，武汉大学出版社．1995．100
[4] Norio Kono. Double points of a Gaussian Sample Path. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 45 (1978): 175-180
[5] LkedaN, WatanabeS. Stochastic Differential Equations and Diffusion Processes. North-Holland Publishing Company New York. 1981
[6] Sheu SJ. Some Estimate of the Transition Density of a Nondegenerate Diffusion Markov Process. Ann, Probab. 1991(2): 538-561
[7] Falconer K. Fractal Geometry, England, 1990
[8] 骆顺龙．张正敏．一类半鞅样本轨道的Fractal性质．应用数学 1996，9(1)：42-45
[9] D. W. Stroock An Introduction to the Theory of Large Deviations. Springer. Verlag, New York, 1984, 81-82.