Abstract :
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.
