Decomposition of a potential theory in elliptic and parabolic subspaces (Q1583993)

Let \(( X,{\mathcal B}) \) be a measurable space and \(V\) a proper kernel on \(X\). Denote by \({\mathcal E}_{V}\) the cone of excessive functions generated by \(V\) satisfying some standard assumptions. \textit{H. Ben Saad} and \textit{K. Janssen} defined in [Math. Anal. 272, 281-289 (1985; Zbl 0571.31008)] parabolicity and ellipticity of the space \(( X,{\mathcal E}_{V}) \). In this paper the authors give a necessary and sufficient condition to decompose the space \(( X,{\mathcal E}_{V}) \) in an ordered and countable family of parabolic or elliptic subspaces \(( X_{i},{\mathcal E}_{V_{i}}) \), where \(X_{i}\) are finely open and form a partition of \(X \), the kernel \(\sum V_{i}\) on \(X=\sum X_{i}\) is subordinate to \(V\) and has a triangular matrix.