Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

Consider the symmetric non-local Dirichlet form $(D,\text{D}(D))$ given by $$D(f,f)={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}(f(x)-f(y){)}^{2}J(x,y)dxdy$$with $\text{D}(D)$ the closure of the set of $C^1$ functions on ${\mathbb{R}}^d$ with compact support under the norm $\sqrt{D_1(f,f)}$, where $D_1(f,f):=D(f,f)+\int f^2(x),dx$and$J(x,y)$isanonnegativesymmetricmeasurablefunctionon$\R^d\times \R^d$\mathrm{.}SupposethatthereisaHuntprocess$(X_t)_{t\ge 0}$on$\R^d$correspondingto$(D,\D(D))$,andthat$(L,\D(L))$ is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigroup $(T_t^V{)}_{t\ge 0}$ generated by $L^V:=L-V$, where $V\ge 0$ is a non-negative locally bounded measurable function such that Lebesgue measure of the set $\{x\in {\mathbb{R}}^d:V(x)\le r\}$ is finite for every $r>0$. By using intrinsic super Poincar'{e} inequalities and establishing an explicit lower bound estimate for the ground state, we present general criteria for the intrinsic ultracontractivity of $(T_t^V{)}_{t\ge 0}$. In particular, if $$J(x,y)\asymp \mathrm{\mid }x-y{\mathrm{\mid }}^{-d-\alpha }{\text{I}}_{\{\mathrm{\mid }x-y\mathrm{\mid }\le 1\}}+e^{-\mathrm{\mid }x-y{\mathrm{\mid }}^{\gamma }}{\text{I}}_{\{\mathrm{\mid }x-y\mathrm{\mid }>1\}}$$ for some $\alpha \in (0,2)$ and $\gamma \in (1,\mathrm{\infty }]$, and the potential function $V(x)=\mathrm{\mid }x{\mathrm{\mid }}^{\theta }$ for some $\theta >0$, then $(T_t^V{)}_{t\ge 0}$ is intrinsically ultracontractive if and only if $\theta >1$. When $\theta >1$, we have the following explicit estimates for the ground state ${\varphi }_1$ $$c_1\exp (-c_2{\theta }^{\frac{\gamma -1}{\gamma }}\mathrm{\mid }x\mathrm{\mid }{\log }^{\frac{\gamma -1}{\gamma }}(1+\mathrm{\mid }x\mathrm{\mid }))\le {\varphi }_1(x)\le c_3\exp (-c_4{\theta }^{\frac{\gamma -1}{\gamma }}\mathrm{\mid }x\mathrm{\mid }{\log }^{\frac{\gamma -1}{\gamma }}(1+\mathrm{\mid }x\mathrm{\mid })),$$ where $c_i>0$ $(i=1,2,3,4)$ are constants. We stress that, our method efficiently applies to the Hunt process $(X_t{)}_{t\ge 0}$ with finite range jumps, and some irregular potential function $V$ such that ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}V(x)\mathrm{\ne }\mathrm{\infty }$.