Decomposition of a theory of potential in elliptic and parabolic subspaces

Let V be a proper kernel on a measurable space (X, B), and E-V the cone of excessive functions generated by V. We give a necessary and sufficient cond ition to decompose the space (X, E-V) in an ordered and countable family of subspaces (X-n, E-Vn), which are elliptic or parabolic. The X-i's are fine ly open and measurable and form a partition of X. The kernel W = Sigma V-i on Sigma X-i is subordinate to V, and has a triangular matrix.