Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

Abstract

Consider the symmetric non-local Dirichlet form $(D,\D(D))$ given by $$D(f,f)={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}(f(x)-f(y){)}^{2}J(x,y)dxdy$$with $\D(D)$ 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 $J(x,y)$ is a nonnegative symmetric measurable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d$. Suppose that there is a Hunt process $(X_t{)}_{t\ge 0}$ on ${\mathbb{R}}^d$ 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 $\{x\in {\mathbb{R}}^d:V(x)\le r\}$ 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 $(T_t^V{)}_{t\ge 0}$. In particular, if $$J(x,y)\asymp|x-y|^{-d-α}\I_{\{|x-y|\le 1\}}+e^{-|x-y|^γ}\I_{\{|x-y|> 1\}}$$ 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 ${\varphi }_1$ $$c_1\exp (-c_2{\theta }^{\frac{\gamma -1}{\gamma }}|x|{\log }^{\frac{\gamma -1}{\gamma }}(1+|x|))\le {\varphi }_1(x)\le c_3\exp (-c_4{\theta }^{\frac{\gamma -1}{\gamma }}|x|{\log }^{\frac{\gamma -1}{\gamma }}(1+|x|)),$$ 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 ${\lim }_{|x|\to \infty }V(x)\ne \infty$.

31 pages

Cited by in corpus (1)

• Intrinsic Contractivity of Feynman-Kac Semigroups for Symmetric Jump Processes with Infinite Range Jumps