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

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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

CLC Number:

31C25

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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