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Unavoidable collections of balls for isotropic Lévy processes

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  • Mimica, Ante
  • Vondraček, Zoran

Abstract

A collection {B¯(xn,rn)}n⩾1 of pairwise disjoint balls in the Euclidean space Rd is said to be avoidable with respect to a transient process X if the process with positive probability escapes to infinity without hitting any ball. In this paper we study sufficient and necessary conditions for avoidability with respect to unimodal isotropic Lévy processes satisfying a certain scaling hypothesis. These conditions are expressed in terms of the characteristic exponent of the process, or alternatively, in terms of the corresponding Green function. We also discuss the same problem for a random collection of balls. The results are generalization of several recent results for the case of Brownian motion.

Suggested Citation

  • Mimica, Ante & Vondraček, Zoran, 2014. "Unavoidable collections of balls for isotropic Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1303-1334.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:3:p:1303-1334
    DOI: 10.1016/j.spa.2013.11.003
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    References listed on IDEAS

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    1. Carroll, Tom & O’Donovan, Julie & Ortega-Cerdà, Joaquim, 2012. "On Lundh’s percolation diffusion," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1988-1997.
    2. Kim, Panki & Song, Renming & Vondraček, Zoran, 2014. "Global uniform boundary Harnack principle with explicit decay rate and its application," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 235-267.
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    Cited by:

    1. Vondraček, Zoran & Wagner, Vanja, 2018. "On purely discontinuous additive functionals of subordinate Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 707-725.
    2. Grzywny, Tomasz & Kwaśnicki, Mateusz, 2018. "Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 1-38.

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