Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

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Consider the symmetric non-local Dirichlet form (D,(D)) given by D(f,f)=∫ᵈ∫ᵈ(f(x)-f(y))² J(x,y)dxdy with (D) the closure of the set of C¹ functions on ᵈ with compact support under the norm √D₁(f,f), where D₁(f,f):=D(f,f)+∫ f²(x)dx and J(x,y) is a nonnegative symmetric measurable function on ᵈ× ᵈ. Suppose that there is a Hunt process (Xₜ)ₜ≥ ₀ on ᵈ corresponding to (D,(D)), and that (L,(L)) is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigroup (Tₜᵛ)ₜ≥ ₀ generated by Lᵛ:=L-V, where V≥ 0 is a non-negative locally bounded measurable function such that Lebesgue measure of the set {x∈ ᵈ: V(x)≤ r} is finite for every r>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, if J(x,y)|x-y|⁻ᵈ⁻α{|ₓ₋y|≤ ₁}+e⁻|ˣ⁻ʸ|γ{|ₓ₋y|> ₁} for some α∈ (0,2) and γ∈(1,∞], and the potential function V(x)=|x|θ for some θ>0, then (Tₜᵛ)ₜ≥ ₀ is intrinsically ultracontractive if and only if θ>1. When θ>1, we have the following explicit estimates for the ground state ϕ₁ c₁(-c₂ θγ-1γ|x| γ-1γ(1+|x|)) ≤ ϕ₁(x) ≤ c₃(-c₄ θγ-1γ|x| γ-1γ(1+|x|)), where cᵢ>0 (i=1,2,3,4) are constants. We stress that, our method efficiently applies to the Hunt process (Xₜ)ₜ ≥ ₀ with finite range jumps, and some irregular potential function V such that |ₓ| → ∞V(x)≠∞.