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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
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