Optimal estimation of shell thickness in Cutland's construction of Wiener measure期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

Authors:	Bang-He Li Ya-Qing Li

Affiliation:	Institute of Systems Science, Academia Sinica, Beijing 100080, People's Republic of China ; Institute of Systems Science, Academia Sinica, Beijing 100080, People's Republic of China

Abstract:	In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure , $begin{displaymath}gamma({x:R_1le \|x\|le R_2})simeq 1,end{displaymath}$ where $R_1=1-(log N)^{frac 12} N^{-frac 12}$ and $R_2=1+MN^{-frac 12}$ with any positive infinite number. We prove here that may be replaced by $1-mN^{-frac 12}$ with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that $gamma({x:\|x\|<1 })simeq gamma ({x:\|x\|>1})simeq frac 12$ . And for the Lebesgue measure , is finite and not infinitesimal iff $r=(2pi e)^{-frac 12}N^{frac 12(1+frac 1N)}e^{frac aN}$ with finite, while for the Lebesgue area of the sphere $S^{N-1}(r)$ , should be $(2pi e)^{-frac 12}N^{frac 12} e^{frac aN}$ .

Keywords:	Shell thickness Wiener measure $*$-finite Euclidean space

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