On Markov processes with decomposable pseudo-differential generators

The paper is devoted to the study of Markov processes in finite-dimensional convex cones (especially R d and ) with a decomposable generator, i.e. with a generator of the form where every A n acts as a multiplication operator by a positive, not necessarily bounded, continuous function a n (x) and where every ψ n generates a Lévy process, i.e. a process with i.i.d. increments in R d . The following problems are discussed: (i) existence and uniqueness of Markov or Feller processes with a given generator, (ii) continuous dependence of the process on the coefficients a n and the starting points, (iii) well posedness of the corresponding martingale problem, (iv) generalized solutions to the Dirichlet problem, (v) regularity of boundary points.