Abstract. Let $X=\{X(t), t\geq0, {\msbm P}^x, x \in G \} $ be the Brownian motion on the Sierpiński gasket $G$. We prove that there exist two positive constants $c$ and $C$ such that for every $x \in G$, ${\msbm P}^x$-a.s. for all $t \in[0, \infty )$, we have $ ct \leq\varphi-m({\rm Gr}(X[0,t]))\leq Ct$, where ${ \rm Gr}X([0,t])=\{(s, X(s)): 0 \leq s \leq t \} $ is the graph set of $X$, $$\varphi(s)=s^{1+ \log3/\log2 - \log3/\log5}(\log\log {1}/{s})^{ \log3/\log5}, s \in(0, {1}/{8}],$$ and $\varphi $-$m$ denotes Hausdorff $\varphi $-measure.

AMS Subject Classification (1991): 60G17, 60J60, 28A78

Keyword(s): Brownian motion on the Sierpiński gasket, Hausdorff measure, graph

Received April 23, 2001, and in revised form October 24, 2001. (Registered under 2871/2009.)