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 R n 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 ( R n ∖ A R n ) < ∞ , we show that X satisfies (H) if and only if the equation A y = − a − ∫ { x ∈ R n ∖ A R n : | x | < 1 } x μ ( d x ) , y ∈ R n , has at least one solution. Finally, we show that if X is a subordinator and satisfies (H) then its drift coefficient must be 0.