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On the path behavior of Markov processes comparable with Lévy processes. (Zum Pfadverhalten von Markovschen Prozessen, die mit Lévy-Prozessen vergleichbar sind.) (German) Zbl 0805.60032
Erlangen-Nürnberg: Univ. Erlangen-Nürnberg, Naturwiss. Fak. 147 S. (1994).
A Markov process $$\{\widetilde {X}_ t\}_{t\geq 0}$$ on $$(\widetilde {\Omega}, \widetilde{\mathcal A}, \widetilde {\mathbb{P}}, \mathbb{R}^ d)$$ is called comparable from above (below) w.r.t. another Markov process $$\{X_ t\}_{t\geq 0}$$ on $$(\Omega,{\mathcal A}, \mathbb{P}, \mathbb{R}^ d)$$ if their one- dimensional transition probabilities are comparable from above (below). Examples are given.
If $$\widetilde{X}_ t$$ is comparable (from above) with $$X_ t$$, various sample path properties – regularity, variation, Hausdorff dimension – of $$X_ t$$ carry over to $$\widetilde{X}_ t$$. If, in particular, $$X_ t$$ is a Lévy process, criteria for the finiteness of the $$\lambda$$- variation and upper and lower bounds for the Hausdorff dimension of $$\widetilde{X}_ t$$ are given. Moreover, the technique of subordination (in the sense of S. Bochner) is studied. Using/developing a functional calculus for $$(C_ 0)$$-semigroups and their generators, a Heinz-Kato type theorem for certain functions of selfadjoint generators is proved.

##### MSC:
 60G17 Sample path properties 60J99 Markov processes 60J75 Jump processes (MSC2010) 47D07 Markov semigroups and applications to diffusion processes 47A60 Functional calculus for linear operators