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On Markov properties of Lévy waves in two dimensions

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  • Dalang, Robert C.
  • Hou, Qiang

Abstract

Markov properties of the solution to the wave equation in two spatial dimensions driven by a Lévy point process are considered. When the velocity of waves is 1, then for domains bounded by a plane, the sharp Markov property is shown to hold if and only if the angle between the plane and the time axis is at least [pi]/4. The sharp Markov property also holds for domains that are bounded polyhedra, because the boundary sigma-field is extremely large. The same is true of the germ-field of the boundary of a bounded open set, and this implies the germ-field Markov property for these sets.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:72:y:1997:i:2:p:265-287
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    References listed on IDEAS

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    1. Dalang, Robert C. & Russo, Francesco, 1988. "A prediction problem for the Brownian sheet," Journal of Multivariate Analysis, Elsevier, vol. 26(1), pages 16-47, July.
    2. Ferrante, Marco & Nualart, David, 1994. "On the Markov property of a stochastic difference equation," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 239-250, August.
    3. Adler, R. J. & Monrad, D. & Scissors, R. H. & Wilson, R., 1983. "Representations, decompositions and sample function continuity of random fields with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 3-30, June.
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    Cited by:

    1. 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.
    2. Balan, Raluca M. & Tudor, Ciprian A., 2010. "The stochastic wave equation with fractional noise: A random field approach," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2468-2494, December.
    3. 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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