ERGODIC PROPERTIES OF CONTINUOUS PARAMETER ADDITIVE PROCESSES

Abstract

Let $\{T_{t}:t\in{\bf R}\}$ be a measure preserving flow in a probability measure space $(\Omega, {\cal A},\mu)$, and $\{F_{t}:t\in {\bf R}\}$ be a family of real-valued measurable functions on $(\Omega,{\cal A},\mu)$ such that $F_{t+s}=F_{t}+F_{s}\circ T_{t}$ (mod $\mu$) for all $t,\, s \in {\bf R}$. In this paper we deduce necessary and sufficient conditions for the existence of a real-valued measurable function $f$ on $\Omega$, with $f\in L_{p}(\Omega,\mu)$ where $0\leq p\leq \infty$, such that $F_{t}=f\circ T_{t}-f$ (mod $\mu$) for all $t\in {\bf R}$. Related results are also obtained. These may be considered to be continuous parameter refinements of the recent discrete parameter results of Alonso, Hong and Obaya concerning additive real coboundary cocycles.