On Ergodic Properties of Some Lévy-Type Processes

In this note we find sufficient conditions for ergodicity of a Lévy-type process with the generator of the corresponding semigroup given by $$\begin{aligned} Lf(x)= & {} a(x)f'(x)\\{} & {} + \int _\mathbb {R}\left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\le 1} \right) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{aligned}$$ L f ( x ) = a ( x ) f ′ ( x ) + ∫ R f ( x + u ) - f ( x ) - ∇ f ( x ) · u 1 | u | ≤ 1 ν ( x , d u ) , f ∈ C ∞ 2 ( R ) . Here $$\nu (x,du)$$ ν ( x , d u ) is a Lévy-type kernel and $$a(\cdot ): \mathbb {R}\rightarrow \mathbb {R}$$ a ( · ) : R → R . We consider the case where the tails of $$\nu (x,\cdot )$$ ν ( x , · ) have polynomial decay, as well as the case where the decay is (sub)-exponential. We use the Foster–Lyapunov approach to prove the results.