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

Abstract

AbstractSuppose 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 AtF be ∑s⩽tF(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 {StF:t⩾0} defined byStFf(x)=Ex(e−AtFf(Xt)) has a density q and that there exist positive constants C1, C2, C3 and C4 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