Unavoidable collections of balls for isotropic Lévy processes

Mimica, Ante; Vondraček, Zoran

doi:10.1016/j.spa.2013.11.003

Cited by 10 publications

(14 citation statements)

References 20 publications

(49 reference statements)

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“…Our first three results provide estimates of the probability of hitting a ball, the potential kernel and the Green function of a ball. All of them improve previously known results: for hitting probabilities, see Lemma 2.5 in [33], Proposition 5.8 in [5] and Lemmas 3.4 2010 Mathematics Subject Classification. Primary: 60J35, 60J50; secondary: 60J75, 31B25.…”

Section: Introductionsupporting

confidence: 83%

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Grzywny

Kwaśnicki

2018

Stochastic Processes and their Applications

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Abstract. In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to O-regular variation at zero and at infinity with lower indices greater than −d − 2. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.Our results are expressed in terms of Pruitt's functions K(r) and L(r), measuring local activity and the amount of large jumps of the Lévy process, respectively.

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Section: Introductionsupporting

confidence: 83%

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

Grzywny

Kwaśnicki

2018

Stochastic Processes and their Applications

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“…For the Brownian semigroup (classical potential theory) and isotropic αstable semigroups (Riesz potentials) we have g(r) = r α−d , α ∈ (0, 2], α < d, and our assumptions are satisfied with R 0 = R 1 = 0. This holds as well for the more general isotropic unimodal Lévy semigroups considered in [11].…”

supporting

confidence: 60%

“…In the more general case of isotropic unimodal Lévy processes, where the characteristic function satisfies a lower scaling condition (and (LD), (UD) hold with R 0 = R 1 = 0), both Theorem 1.2, its converse, and Corollary 1.3 are proven in [11]. We shall use the same method of considering finitely many countable unions of concentric shells, but have to overcome additional difficulties caused by having only a rather weak estimate for the exit distribution of balls (compare [11,Lemma 2.2], going back to [5,Corollary 2], and Proposition 3.7). Nevertheless our proof for Theorem 1.2 can be simpler, since starting with an avoidable union A and an arbitrary δ > 0, we may assume without loss of generality that P 0 [T A < ∞] < δ (using Proposition 3.3 and translation invariance).…”

mentioning

confidence: 97%

“…In Section 2, we shall first take a closer look at the properties (LD) and (UD) and then show that our assumptions cover the isotropic unimodal processes considered in [11] and geometric stable processes. In Section 3, we shall discuss some general potential theory of the semigroup È, where, as in [8], at the beginning (LD) and (UD) are replaced by the weaker properties 1 0 r d−1 g(r) dr < ∞ and lim r→∞ g(r) = 0.…”

mentioning

confidence: 99%

“…The following estimate of the probability for hitting a shell S(r) before leaving a ball B(0, Mr), M large, will be sufficient for us (see [11,Lemma 2.2], going back to [5,Corollary 2], for a much stronger estimate which is used [11]). Proof.…”

mentioning

confidence: 99%

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Unavoidable collections of balls for processes with isotropic unimodal Green function

Hansen¹

2014

Festschrift Masatoshi Fukushima

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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 .

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