  We present a new numerical approach that is able to solve the
multi-dimensional radiative transfer equations in all opacity regimes on a
Lagrangian, unstructured network of characteristics based on a stochastic point
process. Our method reverses the limiting procedure used to derive the transfer
equations, by going back to the original Markov process. Thus, we reduce this
highly complex system of coupled differential equations to a simple
one-dimensional random walk on a graph, which is shown to be computationally
very efficient. Specifically, we use a Delaunay graph, which makes it possible
to combine our scheme with a new smoothed particle hydrodynamics (SPH) variant
proposed by Pelupessy et al.(2003). We show that the results of applying a
two-dimensional implementation of our method with various suitable test cases
agree with the analytical results, and we point out the advantages of using our
method with inhomogeneous point distributions, showing examples in the
progress. Hereafter, we present a supplement to our method, which can be useful
in cases where the medium is optically very thin, and we conclude by stating
some anticipated properties of this method in three dimensions, and announce
future extensions.
