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A general differentiation theorem for \(n\)-dimensional additive processes. (English) Zbl 0603.47008
Let \(T=(T_ t)_{t\in ({\mathbb{R}}_+-\{0\})^ n}\) be an n-parameter strongly continuous semigroup of \(L_ 1\)-contractions. Let \(\{F_ I,I\in {\mathcal J}_ n\}\) be an n-dimensional additive process with respect to T, \(i.e.\)
F\({}_ I\in L_ 1\)
F\({}_{\cup^{k}_{i=1}I_ i}=\sum^{k}_{i=1}F_{I_ i}\) when the \(I_ i\) are disjoint and \(\cup^{k}_{i=1}I_ i\in {\mathcal J}_ n\)
\(\sup_{I\in {\mathcal J}_ n,\lambda_ n(I)>0}\lambda (I)^{-1}\| F_ I\| <+\infty\)
(\({\mathcal J}_ n\) is the set of intervals of \(({\mathbb{R}}_+-\{0\})^ n\) and \(\lambda_ n\) the Lebesgue measure on \({\mathbb{R}}^ n)\). It is then proved that \(\lim_{\epsilon \to 0}\epsilon^{- n}F_{[(0,0,...,0),(\epsilon,...,\epsilon)]}\) exists a.e. This generalizes the case of positive contractions solved by Akcoglu-Del Junco.

MSC:
47A35 Ergodic theory of linear operators
47D03 Groups and semigroups of linear operators
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