  We identify the stochastic processes associated with one-sided fractional
partial differential equations on a bounded domain with various boundary
conditions. This is essential for modelling using spatial fractional
derivatives. We show well-posedness of the associated Cauchy problems in
$C_0(\Omega)$ and $L_1(\Omega)$. In order to do so we develop a new method of
embedding finite state Markov processes into Feller processes and then show
convergence of the respective Feller processes. This also gives a numerical
approximation of the solution. The proof of well-posedness closes a gap in many
numerical algorithm articles approximating solutions to fractional differential
equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence
without checking well-posedness.
