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Strong Feller continuity of Feller processes and semigroups. (English) Zbl 1258.60047
The authors study the strong Feller property for a Markov process and the the corresponding sub-Markov semigroup. The main goal is to establish two equivalent criteria for the strong Feller property: one is described by the local uniform absolute continuity of the transition probabilities \(\operatorname{P}_t(x,dy)\), while the other is based on local Orlicz-ultracontractivity. These two criteria generalize many existing results.
In the last part of the paper, the authors consider Feller processes with the strong Feller property, establish estimates of the first exit times from balls, and investigate continuity of harmonic functions.

MSC:
60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes
60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
60J75 Jump processes (MSC2010)
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