Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

Abstract

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 $\lim_{|x| \to \infty}V(x)\neq\infty$.