  The aim of this paper is two-fold. On one hand, we will study the distorted
Brownian motion on $\mathbb{R}$, i.e. the diffusion process $X$ associated with
a regular and strongly local Dirichlet form obtained by the closure of
$\mathscr{E}(f,g)=\frac{1}{2}\int_\mathbb{R} f'(x)g'(x)\rho(x)dx$ for $f,g\in
C_c^\infty(\mathbb{R})$ on $L^2(\mathbb{R}, \mathfrak{m})$, where
$\mathfrak{m}(dx)=\rho(x)dx$ and $\rho$ is a certain positive function. After
figuring out the irreducible decomposition of $X$, we will present a
characterization of that $X$ becomes a semi-martingale by virtue of so-called
Fukushima's decomposition. Meanwhile, it is also called a general skew Brownian
motion, which turns out to be a weak solution to the stochastic differential
equation with certain $\mu$: \[
  dY_t=dW_t+\int_\mathbb{R}\mu(dz) dL^z_t(Y),\quad (*) \] where $(W_t)_{t\geq
0}$ is a standard Brownian motion and $(L^z_t(Y))_{t\geq 0}$ is the symmetric
semi-martingale local time of the unknown semi-martingale $Y$ at $z$. On the
other hand, the stochastic differential equation (*) will be considered
further. The main purpose is to find the conditions on $\mu$ equivalent to that
there exist general skew Brownian motions being weak solution to (*). Moreover,
the irreducibility and the equivalence in distribution of expected general skew
Brownian motions will be characterized. Finally, several special cases will be
paid particular attention to and we will prove or disprove the pathwise
uniqueness for (*).
