Additive and superadditive local theorems (Q1077073)

Let \(\{T_ t\), \(t>0\}\) be a semigroup of linear operators in \(L_ 1\) with \(\| T_ t\| \leq 1\), which is strongly continuous at \(t>0\) but not necessarily at \(t=0\). Let \(\{F_ t\), \(t>0\}\) be an additive process, i.e. we have \(F_{t+s}=F_ t+T_ tF_ s.\) Assume \(\sup \{\| F_ t\|_ 1/t:\quad 0<t<1\}<\infty.\) It is shown that lim \(F_ t/t\) exists a.e. for \(t\to 0\) ranging through any countable set. This extends a result of the reviewer and \textit{M. Akcoglu} [Math. Z. 163, 199-210 (1978; Zbl 0379.60073)] to the case of nonpositive operators. In the d-parameter case, the author obtains a local ergodic theorem: Let \(\{T_ t:\quad t=(t_ 1,...,t_ d),\quad t_ i>0\}\) be as above, but with a multidimensional parameter. Put \[ A_{\epsilon}f=\epsilon^{- d}\int_{E(\epsilon)}T_{(t_ 1,...,t_ d)}f dt_ 1...dt_ d \] with \(E(\epsilon)=[0,\epsilon]^ d\). It is shown that lim \(A_{\epsilon}f\) exists a.e. for \(\epsilon\) \(\to 0\).