Intrinsic ultracontractivity of Feynman–Kac semigroups for symmetric jump processes

Consider a symmetric non-local Dirichlet form (D,D(D))(D,D(D)) given byD(f,f)=∫Rd∫Rd(f(x)−f(y))2J(x,y)dxdy with D(D)D(D) the closure of the set of C1C1 functions on RdRd with compact support under the norm D1(f,f), where D1(f,f):=D(f,f)+∫f2(x)dx and J(x,y)J(x,y) is a nonnegative symmetric measurable function on Rd×RdRd×Rd. Suppose that there is a Hunt process (Xt)t⩾0(Xt)t⩾0 on RdRd corresponding to (D,D(D))(D,D(D)), and that (L,D(L))(L,D(L)) is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman–Kac semigroup (TtV)t⩾0 generated by LV:=L−VLV:=L−V, where V⩾0V⩾0 is a non-negative locally bounded measurable function such that the Lebesgue measure of the set {x∈Rd:V(x)⩽r}{x∈Rd:V(x)⩽r} is finite for every r>0r>0. By using intrinsic super Poincaré inequalities and establishing an explicit lower bound estimate for the ground state, we present general criteria for the intrinsic ultracontractivity of (TtV)t⩾0. In particular, ifJ(x,y)≍|x−y|−d−α1{|x−y|⩽1}+e−|x−y|γ1{|x−y|>1}J(x,y)≍|x−y|−d−α1{|x−y|⩽1}+e−|x−y|γ1{|x−y|>1} for some α∈(0,2)α∈(0,2) and γ∈(1,∞]γ∈(1,∞], and the potential function V(x)=|x|θV(x)=|x|θ for some θ>0θ>0, then (TtV)t⩾0 is intrinsically ultracontractive if and only if θ>1θ>1. When θ>1θ>1, we have the following explicit estimates for the ground state ϕ1ϕ1c1exp⁡(−c2θγ−1γ|x|logγ−1γ⁡(1+|x|))⩽ϕ1(x)⩽c3exp⁡(−c4θγ−1γ|x|logγ−1γ⁡(1+|x|)), where ci>0ci>0(i=1,2,3,4)(i=1,2,3,4) are constants. We stress that our method efficiently applies to the Hunt process (Xt)t⩾0(Xt)t⩾0 with finite range jumps, and some irregular potential function V   such that lim|x|→∞⁡V(x)≠∞lim|x|→∞⁡V(x)≠∞.