  We develop a general method for extending Markov processes to a larger state
space such that the added points form a polar set. The so obtained extension is
an improvement on the standard trivial extension in which case the process is
made stuck in the added points, and it renders a new technique of constructing
extended solutions to S(P)DEs from all starting points, in such a way that they
are solutions at least after any strictly positive time. Concretely, we adopt
this strategy to study SDEs with singular coefficients on an infinite
dimensional state space (e.g. SPDEs of evolutionary type), for which one often
encounters the situation where not every point in the space is allowed as an
initial condition. The same can happen when constructing solutions of
martingale problems or Markov processes from (generalized) Dirichlet forms, to
which our new technique also applies.
