On estimates of the density of Feynman–Kac semigroups of α-stable-like processes

Suppose that α ∈ (0, 2) and that X is an α-stable-like process on Rd. Let F be a function on Rd belonging to the class Jd,α (see Introduction) and AFt be s t F(Xs−,Xs ), t > 0, a discontinuous additive functional of X. With neither F nor X being symmetric, under certain conditions, we show that the Feynman–Kac semigroup {SF t : t 0} defined by SF t f (x) = Ex e−AFt f (Xt ) has a density q and that there exist positive constants C1, C2, C3 and C4 such that C1e−C2t t−d α 1 ∧ t 1 α |x −y| d+α q(t, x, y) C3eC4t t−d α 1 ∧ t 1 α |x −y| d+α for all (t, x, y) ∈ (0,∞)×Rd ×Rd . © 2008 Published by Elsevier Inc.