  Using elliptic regularity results in weighted spaces, stochastic calculus and
the theory of non-symmetric Dirichlet forms, we first show weak existence of
non-symmetric distorted Brownian motion for any starting point in some domain
$E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict
positivity of the unique continuous version of the density to its invariant
measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e.
a diffusion that typically has an unbounded and discontinuous drift. Once
having shown weak existence, we obtain from a result of \cite{KR} that the
constructed weak solution is indeed strong and weakly as well as pathwise
unique up to its explosion time. As a consequence of our approach, we can use
the theory of Dirichlet forms to prove further properties of the solutions.
More precisely, we obtain new non-explosion criteria for them. We finally
present concrete existence and non-explosion results for non-symmetric
distorted Brownian motion related to a class of Muckenhoupt weights and
corresponding divergence free perturbations.
