数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (1): 141-161.doi: 10.1016/S0252-9602(13)60132-0

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ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

陈振龙   

  1. College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
  • 收稿日期:2012-09-20 修回日期:2013-02-25 出版日期:2014-01-20 发布日期:2014-01-20
  • 基金资助:

    The research was supported by National Natural Science Foundation of China (11371321), Zhejiang Provincial Natural Science Foundation of China (Y6100663), the Key Research Base of Humanities and Social Sciences of Zhejiang Provincial High Education Talents (Statistics of Zhejiang Gongshang University).

ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

 CHEN Zhen-Long   

  1. College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
  • Received:2012-09-20 Revised:2013-02-25 Online:2014-01-20 Published:2014-01-20
  • Supported by:

    The research was supported by National Natural Science Foundation of China (11371321), Zhejiang Provincial Natural Science Foundation of China (Y6100663), the Key Research Base of Humanities and Social Sciences of Zhejiang Provincial High Education Talents (Statistics of Zhejiang Gongshang University).

摘要:

Let X(1) = {X(1)(s), sR+} and X(2) = {X(2)(t), t R+} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X(1) and X(2) are studied. More gener-ally, let E1, E2 ⊆ (0, ∞) and F ⊂ Rd be Borel sets. A necessary condition and a sufficient condition for P{X(1)(E1) ∩X(2)(E2) ∩F 6≠Φ} > 0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1×E2×F in the metric space (R+×R+×Rdρ), where ρ is an unsymmetric metric defined in R+ ×R+ ×Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.

关键词: intersection, diffusion processes, hitting probability, polar set, Hausdorff dimension

Abstract:

Let X(1) = {X(1)(s), sR+} and X(2) = {X(2)(t), t R+} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X(1) and X(2) are studied. More gener-ally, let E1, E2 ⊆ (0, ∞) and F ⊂ Rd be Borel sets. A necessary condition and a sufficient condition for P{X(1)(E1) ∩X(2)(E2) ∩F 6≠Φ} > 0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1×E2×F in the metric space (R+×R+×Rdρ), where ρ is an unsymmetric metric defined in R+ ×R+ ×Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.

Key words: intersection, diffusion processes, hitting probability, polar set, Hausdorff dimension

中图分类号: 

  • 60G15