Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

Consider the symmetric non-local Dirichlet form (D,\D(D)) given by $$D(f,f)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)^2 J(x,y)\,dx\,dy$$with \D(D) the closure of the set of $C^1$ functions on $\R^d$ 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 $J(x,y)$ is a nonnegative symmetric measurable function on $\R^d\times \R^d$. Suppose that there is a Hunt process $(X_t)_{t\ge 0}$ on $\R^d$ corresponding to (D,\D(D)), and that (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 $\{x\in \R^d: V(x)\le r\}$ 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|x-y|^{-d-\alpha}\I_{\{|x-y|\le 1\}}+e^{-|x-y|^\gamma}\I_{\{|x-y|> 1\}} for some $\alpha \in (0,2)$ and $\gamma\in(1,\infty]$, and the potential function $V(x)=|x|^\theta$ 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 $\phi_1$ $$c_1\exp\Big(-c_2 \theta^{\frac{\gamma-1}{\gamma}}|x| \log^{\frac{\gamma-1}{\gamma}}(1+|x|)\Big) \le \phi_1(x) \le c_3\exp\Big(-c_4 \theta^{\frac{\gamma-1}{\gamma}}|x| \log^{\frac{\gamma-1}{\gamma}}(1+|x|)\Big) ,$$ 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 $\lim_{|x| \to \infty}V(x)\neq\infty$.Comment: 31 page

Perturbations of Functional Inequalities for L\'evy Type Dirichlet Forms

Perturbations of super Poincar\'e and weak Poincar\'e inequalities for L\'evy type Dirichlet forms are studied. When the range of jumps is finite our results are natural extensions to the corresponding ones derived earlier for diffusion processes; and we show that the study for the situation with infinite range of jumps is essentially different. Some examples are presented to illustrate the optimality of our results