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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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