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Recurrent lines in two-parameter isotropic stable Lévy sheets

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  • Dalang, Robert C.
  • Khoshnevisan, Davar

Abstract

It is well known that an -valued isotropic [alpha]-stable Lévy process is (neighborhood-)recurrent if and only if d[less-than-or-equals, slant][alpha]. Given an -valued two-parameter isotropic [alpha]-stable Lévy sheet {X(s,t)}s,t[greater-or-equal, slanted]0, this is equivalent to saying that for any fixed s[set membership, variant][1,2], P{t|->X(s,t) is recurrent}=0 if d>[alpha] and =1 otherwise. We prove here that P{[there exists]s[set membership, variant][1,2]: t|->X(s,t) is recurrent}=0 if d>2[alpha] and =1 otherwise. Moreover, for d[set membership, variant]([alpha],2[alpha]], the collection of all times s at which t|->X(s,t) is recurrent is a random set of Hausdorff dimension 2-d/[alpha] that is dense in , a.s. When [alpha]=2, X is the two-parameter Brownian sheet, and our results extend those of Fukushima and Kôno.

Suggested Citation

  • Dalang, Robert C. & Khoshnevisan, Davar, 2004. "Recurrent lines in two-parameter isotropic stable Lévy sheets," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 81-107, November.
    • Handle: RePEc:eee:spapps:v:114:y:2004:i:1:p:81-107
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    References listed on IDEAS

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    1. Bass, Richard F. & Pyke, Ronald, 1987. "A central limit theorem for D(A)-valued processes," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 109-131, February.
    2. Dalang, Robert C. & Hou, Qiang, 1997. "On Markov properties of Lévy waves in two dimensions," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 265-287, December.
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    Cited by:

    1. Herbin, Erick & Merzbach, Ely, 2013. "The set-indexed Lévy process: Stationarity, Markov and sample paths properties," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1638-1670.

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    More about this item

    Keywords

    Stable sheets Recurrence;

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