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