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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.

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