Intrinsic Ultracontractivity of Non-local Dirichlet forms on Unbounded Open Sets

2017/06/25 by Xin Chen, Chen, Xin, Panki Kim +3
Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #advanced mathematical theories #math.FA #math.PR

paper · pdf · doi:10.48550/arxiv.1706.08031

47 pages

arxiv created 2017/06/25 · arxiv updated 2017/06/27

Abstract

In this paper we consider a large class of symmetric Markov processes X=(X_t)_t≥0 on \R^d generated by non-local Dirichlet forms, which include jump processes with small jumps of α-stable-like type and with large jumps of super-exponential decay. Let D⊂ \R^d be an open (not necessarily bounded and connected) set, and X^D=(X_t^D)_t≥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≥0 of X^D. 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.

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