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Current Trends in Random Walks on Random Lattices

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  • Jewgeni H. Dshalalow

    (Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32940, USA)

  • Ryan T. White

    (Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32940, USA)

Abstract

In a classical random walk model, a walker moves through a deterministic d -dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

Suggested Citation

  • Jewgeni H. Dshalalow & Ryan T. White, 2021. "Current Trends in Random Walks on Random Lattices," Mathematics, MDPI, vol. 9(10), pages 1-38, May.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1148-:d:557834
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    References listed on IDEAS

    as
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    4. Jewgeni H. Dshalalow, 1994. "First excess levels of vector processes," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-8, January.
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    6. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    7. Takashi Odagaki & Keisuke Kasuya, 2017. "Alzheimer random walk," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(9), pages 1-5, September.
    8. Jewgeni H. Dshalalow & Jay Yellen, 1996. "Bulk input queues with quorum and multiple vacations," Mathematical Problems in Engineering, Hindawi, vol. 2, pages 1-12, January.
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    11. Jewgeni H. Dshalalow & Jean-Baptiste Bacot, 2001. "On functionals of a marked Poisson process observed by a renewal process," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 26, pages 1-10, January.
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