We have the following setting.
Let $D$ be a bounded domain. We consider the Dirichlet form $(\mathscr{E}, \mathscr{F})$ on $L^2(D,dx)$ with $\mathscr{F}=H^1(D)$ and
$\mathscr{E}(u,v)=\dfrac{1}{2}\int_D\sum^d_{i,j=1}\frac{\partial}{\partial x^j}a^{ij}\frac{\partial}{\partial x^{i}}dx$
with the associated diffusion process $(X_t)_{t \geq 0}$.
If $\partial D$ and $a=(a^{ij})$ are smooth, it follows that the coordinate processes $X^i=(X^i_t)_{t \geq 0}$ are continuous semi-martingales.
Why does this last conclusion hold?