Unavoidable collections of balls for processes with isotropic unimodal Green function

Abstract

Let us suppose that we have a right continuous Markov semigroup on Ê d , d ≥ 1, such that its potential kernel is given by convolution with a function G 0 = g(| · |), where g is decreasing, has a mild lower decay property at zero, and a very weak decay property at infinity. This captures not only the Brownian semigroup (classical potential theory) and isotropic α-stable semigroups (Riesz potentials), but also more general isotropic Lévy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes.There always exists a corresponding Hunt process. A subset A of Ê d is called unavoidable, if the process hits A with probability 1, wherever it starts. It is known that, for any locally finite union of pairwise disjoint balls B(z, r z ), z ∈ Z, which is unavoidable, z∈Z g(|z|)/g(r z ) = ∞. The converse is proven assuming, in addition, that, for some ε > 0, |z − z ′ | ≥ ε|z|(g(|z|)/g(r z )) 1/d , whenever z, z ′ ∈ Z, z = z ′ . It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects Z, and r z = g(φ(|z|)), where φ is a decreasing function.The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondraček. Introduction and main resultsLet È = (P t ) t>0 be a right continuous Markov semigroup on Ê d , d ≥ 1, such that its potential kernel V 0 := ∞ 0 P t dt is given by convolution with a È-excessive functionwhere g is a decreasing function on [0, ∞) such that 0 < g < ∞ on (0, ∞), lim r→0 g(r) = g(0) = ∞ and the following holds:(LD) Lower decay property: There are R 0 ≥ 0 and C G ≥ 1 such that (1.1) d r 0 s d−1 g(s) ds ≤ C G r d g(r), for all r > R 0 .

Liouville Property, Wiener’s Test and Unavoidable Sets for Hunt Processes

“…1. We note that Assumptions 1.4 and 1.8 are satisfied by rather general isotropic Lévy processes (often with R 0 = 0; see [4] and [7] for details).…”

“…so that, in particular, (1.14) cap B(x, r) ≥ c −1 0 g(r) −1 (for many Lévy processes, the Lebesgue measure will have this property; see [7]). COROLLARY 1.13.…”

“…The next simple result on comparison of potentials (cf. [7]) will be sufficient for us (see the proof of [11,Theorem 5.3] for a much more delicate version; cf. also the proof of [1,Theorem 3]).…”

Hunt’s hypothesis (H) and triangle property of the Green function

“…Then (R d , E P ) is a balayage space such that G(x, y) := G 0 (x − y) satisfies Assumption A as well as the following Assumption B (see [9,Section 6] and [6]; cf. [7] for more general Lévy processes).…”

“…The purpose of this short paper is to show that in the settings considered in [6,7,9,10] Hunt's hypothesis (H) holds, that is, semipolar sets are polar provided the underlying space X is an abelian group and the set W of positive hyperharmonic functions on X (the set of excessive functions of a corresponding Hunt process) is invariant under the group action. The essential property we use is a local triangle property of a Green function for (X, W).…”