Null flows, positive flows and the structure of stationary symmetric stable processes

Abstract

This paper elucidates the connection between stationary symmetric α-stable processes with 0<α<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.