数学物理学报 ›› 2010, Vol. 30 ›› Issue (6): 1485-1494.

• 论文 • 上一篇    下一篇

布朗运动可加泛函渐近性的一些新结果

陈传钟1|韩新方2|马丽3   

  1. 1.海南师范大学数学与统计学院 海口 571158;2.中南大学数学科学与计算技术学院 长沙 410075|中科院数学与系统科学研究院应用数学所 |北京 100190;3. Department of Mathematics and Statistics, Concordia University, Montreal H4B1R6, Canada
  • 收稿日期:2008-11-30 修回日期:2009-09-15 出版日期:2010-12-25 发布日期:2010-12-25
  • 基金资助:

    国家自然科学基金(10961012)、海南省自然科学基金(80529)和海南师范大学博士基金资助

Some New Results about Asymptotic Properties of Additive Functionals of Brownian Motion

 CHEN Chuan-Zhong1, HAN Xin-Fang2, MA Li3   

  1. 1.Department of Mathematics and Statistics, Hainan Normal University, Haikou 571158;2.School of Mathematical Science and Computing
    Technology, Central South University, Changsha 410075|Institute of Applied Mathematics, AMSS, CAS, Beijing 100190;3.Department of Mathematics and Statistics, Concordia University, Montreal H4B 1R6, Canada
  • Received:2008-11-30 Revised:2009-09-15 Online:2010-12-25 Published:2010-12-25
  • Supported by:

    国家自然科学基金(10961012)、海南省自然科学基金(80529)和海南师范大学博士基金资助

摘要:

B=(Ω,F,(Ft)t0,(Bt)t0,(Px)x\inRd)L2(Rd,m)上经典的布朗运动, (E,D(E))为其联系的对称狄氏型. 设uD(E), ˜u(Bt)˜u(B0)=Mut+Nut˜u(Bt)的Fukushima分解. 该文主要研究由上鞅可乘泛函Lut:=eMut12Mut(Bt)t0进行变换所得到的新过程(ˆBt)t0 的一些性质; 同时还研究了由Nut产生的布朗运动可加泛函渐近性问题, 并得到了新的结果:  如果 u有界, uKd1, 且Lut 是鞅, ||E.(eMut)||q<, 那么对任意的xRd
$$
     \lim_{t\rightarrow \infty}\frac{1}{t}
     \log E_{x}(e^{N^{u}_{t}})
     =-\inf_{{f\in {{\cal D}}({\cal E})_{b}}\atop{\|   f \| _{L^{2}(R^{d},m)}=1}}({{\cal E}}(f,f)+{{\cal E}}(f^{2},u)).

关键词: 狄氏型, Fukushima分解, 布朗运动, 转移密度函数, 渐近性

Abstract:

Let B=(Ω,F,(Ft)t0,(Bt)t0,(Px)x\inRd) be the classical Brownian motion on L2(Rd,m), which is associated with a symmetric Dirichlet form (E,D(E)). For uD(E)˜u(Bt)˜u(B0)=Mut+Nut is Fukushima decomposition, where ˜u is a quasi-continuous version of uMut the martingale part and Nut  the zero energy part. In this paper, the authors first study transformed process ˆB of B, which is determined by the supermartingale Lut:=eMut12Mut, they get some properties of its transition semigroup; Then, they study the asymptotic properties of Nut, they get that if Lut  is a martingale, u is bounded and
uKd1, ||E.(eMut)||q<, then for every xRdlimt1tlogEx(eNut)=inffD(E)bfL2(Rd,m)=1(E(f,f)+E(f2,u)),

where
 D(E)b=D(E)L(Rd,m).

Key words: Dirichlet form, Fukushima decomposition, Brownian motion, Transition density function Asymptotic property

中图分类号: 

  • 31C25