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Approximation of Brownian Motion on Simple Graphs

Author

Listed:
  • George A. Anastassiou

    (Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA)

  • Dimitra Kouloumpou

    (Section of Mathematics, Hellenic Naval Academy, 18539 Piraeus, Greece)

Abstract

This article is based on chapters 9 and 19 of the new neural network approximation monograph written by the first author. We use the approximation properties coming from the parametrized and deformed neural networks based on the parametrized error and q -deformed and β -parametrized half-hyperbolic tangent activation functions. So, we implement a univariate theory on a compact interval that is ordinary and fractional. The result is the quantitative approximation of Brownian motion on simple graphs: in particular over a system S of semiaxes emanating from a common origin radially arranged and a particle moving randomly on S . We produce a large variety of Jackson-type inequalities, calculating the degree of approximation of the engaged neural network operators to a general expectation function of this kind of Brownian motion. We finish with a detailed list of approximation applications related to the expectation of important functions of this Brownian motion. The differentiability of our functions is taken into account, producing higher speeds of approximation.

Suggested Citation

  • George A. Anastassiou & Dimitra Kouloumpou, 2023. "Approximation of Brownian Motion on Simple Graphs," Mathematics, MDPI, vol. 11(20), pages 1-27, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4329-:d:1262136
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    References listed on IDEAS

    as
    1. Vassilis G. Papanicolaou & Effie G. Papageorgiou & Dimitris C. Lepipas, 2012. "Random Motion on Simple Graphs," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 285-297, June.
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