Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes

In this paper, Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes are revisited. Let X be a Lévy process on Rn with Lévy–Khintchine exponent (a,A,μ). First, we show that if A is non-degenerate then X satisfies (H). Second, under the assumption that μ(Rn∖ARn)<∞, we show that X satisfies (H) if and only if the equation Ay=−a−∫{x∈Rn∖ARn:|x|<1}xμ(dx),y∈Rn, has at least one solution. Finally, we show that if X is a subordinator and satisfies (H) then its drift coefficient must be 0.