布朗运动可加泛函渐近性的一些新结果

布朗运动可加泛函渐近性的一些新结果

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

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

陈传钟¹|韩新方²|马丽³

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

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

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：

设 $\textbf{B=}(\Omega,{{\cal F}},({{\cal F}}_{t})_{t\geq0},(B_{t})_{t\geq0},(P_{x})_{x\inR^{d}})$ 为 $L^{2}(R^{d},m)$ 上经典的布朗运动, $({{\cal E}},{{\cal D}}({{\cal E}}))$ 为其联系的对称狄氏型. 设 $u\in{{\cal D}}({{\cal E}})$ , $\tilde{u}(B_{t})-\tilde{u}(B_{0})=M^{u}_{t}+N^{u}_{t}$ 为 $\tilde{u}(B_{t})$ 的Fukushima分解. 该文主要研究由上鞅可乘泛函 $L^{-u}_{t}:=e^{M^{-u}_{t}-\frac{1}{2}\langle M^{-u}\rangle_{t}}$ 对 $(B_{t})_{t\geq0}$ 进行变换所得到的新过程 $(\widehat{B}_{t})_{t\geq0}$ 的一些性质; 同时还研究了由 $N^{u}_{t}$ 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, $\nabla u\in K_{d-1}$ , 且 $L^{-u}_{t}$ 是鞅, $||E_{.}(e^{M^{-u}_{t}})||_{q}<\infty$ , 那么对任意的 $x\in R^{d}$ 有
$$
     \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 $\textbf{B=}(\Omega,{{\cal F}},({{\cal F}}_{t})_{t\geq0},(B_{t})_{t\geq0},(P_{x})_{x\inR^{d}})$ be the classical Brownian motion on $L^{2}(R^{d},m)$ , which is associated with a symmetric Dirichlet form $({{\cal E}},{{\cal D}}({{\cal E}}))$ . For $u\in{{\cal D}}({{\cal E}})$ , $\tilde{u}(B_{t})-\tilde{u}(B_{0})=M^{u}_{t}+N^{u}_{t}$ is Fukushima decomposition, where $\tilde{u}$ is a quasi-continuous version of $u$ , $M^{u}_{t}$ the martingale part and $N^{u}_{t}$ the zero energy part. In this paper, the authors first study transformed process $\widehat{B}$ of B, which is determined by the supermartingale $L^{-u}_{t}:=e^{M^{-u}_{t}-\frac{1}{2}\langle M^{-u}\rangle_{t}}$ , they get some properties of its transition semigroup; Then, they study the asymptotic properties of $N^{u}_{t}$ , they get that if $L^{-u}_{t}$ is a martingale, $u$ is bounded and
$\nabla u\in K_{d-1}$ , $||E_{.}(e^{M^{-u}_{t}})||_{q}<\infty$ , then for every $x\in R^{d}$ ,

\begin{array}{rcl} lim_{t \to \infty} \frac{1}{t} \log E_{x} (e^{N_{t}^{u}}) = - inf_{\binom{f \in D (E)_{b}}{‖ f ‖_{L^{2} (R^{d}, m)} = 1}} (E (f, f) + E (f^{2}, u)), \end{array}

$\begin{eqnarray*} \lim_{t\rightarrow \infty}\frac{1}{t} \log E_{x}(e^{N^{u}_{t}}) =-\inf_{{f\in {{\cal D}}({\cal E})_{b}}\atop{\| f \| _{L^{2}(R^{d},m)}=1}}({{\cal E}}(f,f)+{{\cal E}}(f^{2},u)), \end{eqnarray*}$ where

D (E)_{b} = D (E) \cap L^{\infty} (R^{d}, m)

${{\cal D}}({{\cal E}})_{b}={{\cal D}}({{\cal E}})\cap L^{\infty}(R^{d},m)$ .

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

中图分类号:

31C25

陈传钟, 韩新方, 马丽. 布朗运动可加泛函渐近性的一些新结果[J]. 数学物理学报, 2010, 30(6): 1485-1494.

CHEN Chuan-Zhong, HAN Xin-Fang, MA Li. Some New Results about Asymptotic Properties of Additive Functionals of Brownian Motion[J]. Acta mathematica scientia,Series A, 2010, 30(6): 1485-1494.

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