The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains
The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains
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DOI：
10.1007/s10959-010-0306-7
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发表时间：
2011-12
影响因子：
0.8
通讯作者：
Dawei Lu;Lixin Song
中科院分区：
文献类型：
--
作者：
Dawei Lu;Lixin Song
Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, $D_{\min}=\{(x,y_{1},y_{2}):\|x\|< \min\{(y_{1}+1)^{1/p_{1}},(y_{2}+1)^{1/p_{2}}\}\}$ and $D_{\max}=\{(x,y_{1},y_{2}):\|x\|<\max\{(y_{1}+1)^{1/p_{1}},\allowbreak(y_{2}+1)^{1/p_{2}}\}\}$ inRd+2,d≥1, respectively, where ‖⋅‖ is the Euclidean norm inRd,y1,y2≥−1, andp1,p2>1. Letanddenote the first times the Brownian motion exits fromDminandDmax. Estimates with exact constants for the asymptotics ofandare given ast→∞, depending on the relationship betweenp1andp2, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.
Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, $D_{\min}=\{(x,y_{1},y_{2}):\|x\|< \min\{(y_{1}+1)^{1/p_{1}},(y_{2}+1)^{1/p_{2}}\}\}$ and $D_{\max}=\{(x,y_{1},y_{2}):\|x\|<\max\{(y_{1}+1)^{1/p_{1}},\allowbreak(y_{2}+1)^{1/p_{2}}\}\}$ inRd+2,d≥1, respectively, where ‖⋅‖ is the Euclidean norm inRd,y1,y2≥−1, andp1,p2>1. Letanddenote the first times the Brownian motion exits fromDminandDmax. Estimates with exact constants for the asymptotics ofandare given ast→∞, depending on the relationship betweenp1andp2, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.