On recurrence and transience of some Lévy-type processes in ℝ

In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in R \mathbb {R} , whose generator defined on the test functions is of the form L f ( x ) = ∫ R ( f ( x + u ) − f ( x ) − ∇ f ( x ) ⋅ u 1 | u | ≤ 1 ) ν ( x , d u ) , f ∈ C ∞ 2 ( R ) . \begin{equation*} Lf(x) =\int _{\mathbb {R}} \left ( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\leq 1} \right ) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{equation*} Here ν ( x , d u ) \nu (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.