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Telegraph Process with Elastic Boundary at the Origin

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Abstract

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards, with probability 1−α. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Antonio Di Crescenzo & Barbara Martinucci

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Shelemyahu Zacks

Authors
  1. Antonio Di Crescenzo
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  2. Barbara Martinucci
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  3. Shelemyahu Zacks
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Correspondence to Barbara Martinucci.

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Research partially supported by the group GNCS of INdAM.

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Di Crescenzo, A., Martinucci, B. & Zacks, S. Telegraph Process with Elastic Boundary at the Origin. Methodol Comput Appl Probab 20, 333–352 (2018). https://doi.org/10.1007/s11009-017-9549-4

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  • DOI: https://doi.org/10.1007/s11009-017-9549-4

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