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Density in small time for Lévy processes. (English) Zbl 0899.60065

Consider a real-valued Lévy process with density \(p(x, t)\). The author proves that according to the limiting behaviour of \(\log(p(x, t))\) as \(t\to 0\) for different \(x\), the real line can be divided into three subsets: (i) the set of points that the process can reach with a finite number of jumps; (ii) the set of points that the process can reach with an infinite number of jumps; (iii) the set of points that the process cannot reach with only jumps (this does not mean that these points are inaccessible for the process). Different examples illustrating the result are given.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60E07 Infinitely divisible distributions; stable distributions
60J65 Brownian motion
60J75 Jump processes (MSC2010)
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References:

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