Long-time behavior of stable-like processes

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol p ( x , ξ ) = − i β ( x ) ξ + γ ( x ) | ξ | α ( x ) , where α ( x ) ∈ ( 0 , 2 ) , β ( x ) ∈ R and γ ( x ) ∈ ( 0 , ∞ ) . More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function α ( x ) , the drift function β ( x ) and the scaling function γ ( x ) . Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable Lévy processes with the index of stability α ≠ 1 .