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

Suppose that α∈(0,2)α∈(0,2) and that X is an α  -stable-like process on RdRd. Let F   be a function on RdRd belonging to the class Jd,αJd,α (see Introduction) and AtF be ∑s⩽tF(Xs−,Xs)∑s⩽tF(Xs−,Xs), t>0t>0, a discontinuous additive functional of X. With neither F nor X   being symmetric, under certain conditions, we show that the Feynman–Kac semigroup {StF:t⩾0} defined byStFf(x)=Ex(e−AtFf(Xt)) has a density q   and that there exist positive constants C1C1, C2C2, C3C3 and C4C4 such thatC1e−C2tt−dα(1∧t1α|x−y|)d+α⩽q(t,x,y)⩽C3eC4tt−dα(1∧t1α|x−y|)d+α for all (t,x,y)∈(0,∞)×Rd×Rd(t,x,y)∈(0,∞)×Rd×Rd.