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.