Theory of Probability and Mathematical Statistics
On recurrence and transience of some Lévy-type processes in R
Victoria Knopova
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Abstract: In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in R, whose generator defined on the test functions is of the form
Lf(x)=∫R(f(x+u)-f(x)-f(x)uI|u|≤1)v(x,du), f∈C∞2(R).
Here v(x,du) is a Lévy-type kernel, whose tails are either extended regularly varying or decaying fast enough. For the proof the Foster–Lyapunov approach is used.
Keywords: Recurrence, transience, Lévy-type process, Foster–Lyapunov criteria, Lyapunov function
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