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On Lévy’s Brownian motion and white noise space on the circle

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  • Huang, Chunfeng
  • Li, Ao

Abstract

In this article, we show that the Brownian motion on the circle constructed by Lévy (1959) is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the other half-circle, and is degenerated in the sense of Minlos (1959). This raises the question of what the white noise is on the circle. We then formally define the white noise space and its associated Brownian bridge on the circle.

Suggested Citation

  • Huang, Chunfeng & Li, Ao, 2021. "On Lévy’s Brownian motion and white noise space on the circle," Statistics & Probability Letters, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:stapro:v:171:y:2021:i:c:s0167715221000031
    DOI: 10.1016/j.spl.2021.109041
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    References listed on IDEAS

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    1. Wood, Andrew T. A., 1995. "When is a truncated covariance function on the line a covariance function on the circle?," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 157-164, August.
    2. Giacomo Aletti & Matteo Ruffini, 2017. "Is the Brownian bridge a good noise model on the boundary of a circle?," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 389-416, April.
    3. Asger Hobolth & Jan Pedersen & Eva Jensen, 2003. "A continuous parametric shape model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(2), pages 227-242, June.
    4. Finn Lindgren & Håvard Rue & Johan Lindström, 2011. "An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 423-498, September.
    5. Huang, Chunfeng & Zhang, Haimeng & Robeson, Scott M., 2016. "Intrinsic random functions and universal kriging on the circle," Statistics & Probability Letters, Elsevier, vol. 108(C), pages 33-39.
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