L^p‐Liouville property for non‐local operators

Abstract

The L p -Liouville property of a non-local operator A is investigated via the associated Dirichlet form (E, F ). We will show that any non-negative continuous E-subharmonic function f ∈ F loc ∩ L p are constant under a quite mild assumption on the kernel of E if p ≥ 2. On the contrary, if 1 < p < 2, we need an additional assumption: either, the kernel has compact support; or f is Hölder continuous

On stochastic completeness of jump processes

“…Proof The proof is taken from [26, Proposition 2.2] and [27]. By Lemma 3.5 vw ∈ F so that E (u, vw) makes sense.…”

“…The integral (28) converges because w is bounded while u, v ∈ F . The integral (27) converges because the integrand is bounded by (x, y) and, by the Cauchy-Schwarz inequality and (21),…”

On $L^p$ Liouville theorems for Dirichlet forms

“…The first result is an analogue of Yau's L p -Liouville type theorem [16]. For some nonlocal operators on Euclidean space a related result is contained in [12]. Speaking about (sub)harmonic functions we refer here to weakly (sub)harmonic functions, see Definition 3.4.…”