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Random Walks with Invariant Loop Probabilities: Stereographic Random Walks.

Miquel Montero1,2

  • 1Departament de Física de la Matèria Condensada, Universitat de Barcelona (UB), Martí i Franquès 1, E-08028 Barcelona, Spain.

Entropy (Basel, Switzerland)
|July 2, 2021
PubMed
Summary
This summary is machine-generated.

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This study introduces a family of Markov processes called random walks with invariant loop probabilities. These processes, including the simple random walk, are derived from geometric principles and applied to a circle with reflexing boundaries.

Area of Science:

  • Probability theory
  • Stochastic processes
  • Geometric probability

Background:

  • Markov processes are fundamental in modeling random phenomena.
  • Random walks are a key type of Markov process with broad applications.
  • Invariant loop probabilities define a specific class of random walks with unique properties.

Purpose of the Study:

  • To introduce and characterize random walks with invariant loop probabilities.
  • To explore the geometric origins of these processes via stereographic projection.
  • To analyze the elliptic case: random walks on a circle with reflexing boundaries.

Main Methods:

  • Development of Markov processes with site-dependent transition probabilities.
  • Application of geometric considerations, specifically stereographic projection.
Keywords:
elliptic geometryheterogeneous mediumhyperbolic geometryrandom walksurvival analysis

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  • Analysis of random walks on a circular domain with reflexing boundaries.

    • Main Results:

    • A unified framework for a wide family of Markov processes, including simple random walks.
    • Demonstration of geometric origins for these random walks.
    • Characterization of the elliptic case, focusing on circular random walks.

    Conclusions:

    • Random walks with invariant loop probabilities offer a rich framework for stochastic modeling.
    • Geometric insights provide a novel perspective on the construction of these processes.
    • The elliptic case provides a specific, tractable model for further study.