Intrinsic Ultracontractivity and Ground State Estimates of Non-local Dirichlet forms on Unbounded Open Sets

Abstract

In this paper we consider a large class of symmetric Markov processes $${X=(X_t)_{t\ge0}}$$ on $${\mathbb{R}^d}$$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $${\alpha}$$ -stable-like type and with large jumps of super-exponential decay. Let $${D\subset \mathbb{R}^d}$$ be an open (not necessarily bounded and connected) set, and $${X^D=(X_t^D)_{t \ge 0}}$$ be the killed process of X on exiting D. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $${(P^{D}_t)_{t\ge0}}$$ of XD. When D is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of X and the reference function of the horn-shaped region D.