Ergodic properties of continuous parameter additive processes (Q1417164)

Let \(\{T_t,\, t\in\mathbb{R}\}\) be a measure preserving flow in a probability space \((\Omega,{\mathcal A},\mu)\). An additive process \(\{F_t,\, t\in\mathbb{R}\}\) is a family of real-valued measurable functions such that \(F_{t+s}(\omega)= F_t(\omega)+ F_s(T_t\omega)\) holds a.e. for all \(t,s\in\mathbb{R}\), and the map \(t\to F_t\) is continuous in the space \(L_0\) of real valued measurable functions on \(\Omega\). The main aim of the paper is to deduce necessary and sufficient conditions for a representation \(F_t= f\circ T_t- f\) with \(f\in L_p\) \((0\leq p\leq\infty)\). To this end, the author introduces a skew product transformation in \(\Omega\times\mathbb{R}\) by \(\vartheta_t(\omega,x)= (T_t \omega,x+ F_t(\omega))\). The existence of a function \(f\in L_0\) such that \(F_t= f\circ T_t- f\) is equivalent to the existence of a finite invariant measure equivalent to \(\mu\otimes(dx/(\pi(1 +x^2)))\). An equivalent condition is that for a.e. \(\omega\) there exists \(N\geq 1\) such that \[ \limsup_{b\to\infty}{1\over b} \int^b_0 \mathbf{1}_{(-N,N)}(F_t(\omega))\,dt> 0. \] Also, the relationship between such conditions and the validity of the pointwise ergodic theorem for \(\{\vartheta_t\}\) is studied. There are further related results.