Skip to main content
SpringerLink
Log in

Simulation example of a black noise

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Tsirelson and Vershik (In Rev Math Phys 10(1):81–145, 1998) introduced the notion of a mathematical noise, which possesses completely opposite properties to those of a white noise. Afterward, Tsirelson (In Lectures on probability theory and statistics, Lecture Notes in Math., 1840, Springer, Berlin, pp 1–106, 2004) called this noise: ‘black noise.’ In this paper, we provide a method to simulate black noise using a modified Bayesian convolutional neural network. Then we study the behavior of black noise both numerically and visually.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. A pair \((p, {\textbf{D}}_{p})\) is called a point function (which we denote by p again) on a measurable space \({\textbf{X}}\) if \({\textbf{D}}_{p} \subset {\mathbb {R}}\) is countable and \(p: {\textbf{D}}_{p} \rightarrow {\textbf{X}}\). We denote by \(\Pi _{{\textbf{X}}}\) the set of all point functions on \({\textbf{X}}\). Then the mapping \( \Pi _{{\textbf{X}}} \ni p \mapsto N_{p} ( \textrm{d}t \textrm{d}x ):= \# \{ \textrm{d}t \cap {\textbf{D}}_{p}: p(x) \in \textrm{d}x \} \) induces a measurable structure on \(\Pi _{{\textbf{X}}}\). A \(\Pi _{{\textbf{X}}}\)-valued random variable p is called a stationary Poisson point process on a \(\sigma \)-finite measure space \(({\textbf{X}}, n)\) with the characteristic measure n if \( N_{p} ( \textrm{d}t \textrm{d}x ) \) is a Poisson random measure with intensity \( \textrm{d}t \otimes n ( \textrm{d}x ) \).

  2. We refer the reader to the original article, [19], in regards to the exact meaning of ‘uniform’ in this sentence.

References

  1. Akahori, J., Izumi, M., Watanabe, S.: Noises, stochastic flows and \(E_{0}\)-semigroups. In: Selected papers on probability and statistics (Papers translated from the Japanese, originally published in Sūgaku), Am. Math. Soc. Transl. Ser. 2, 227, Am. Math. Soc., Providence, RI, 2009, pp. 1–23. https://doi.org/10.1090/trans2/227

  2. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950). https://doi.org/10.1090/S0002-9947-1950-0051437-7

    Article  MathSciNet  Google Scholar 

  3. Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90, 5–43 (1999)

    Article  MathSciNet  Google Scholar 

  4. Ellis, T., Feldheim, O.N.: The Brownian web is a two-dimensional black noise. Ann. Inst. Henri Poincaré Probab. Stat. 52(1), 162–172 (2016)

    Article  MathSciNet  Google Scholar 

  5. Itô, K.: Stochastic Integral. Proc. Imperial Acad. Tokyo 20, 519–524 (1944)

    MathSciNet  Google Scholar 

  6. Itô, K.: On stochastic differential equations. Memoirs Am. Math. Soc. 4, 1–51 (1951)

    Google Scholar 

  7. Kalai, G.: Is the Universe Noise-Sensitive?. arXiv preprint (2007). arXiv:hep-th/0703092

  8. Kalai, G.: Noise Sensitivity–The case of Percolation. presented in Hebrew University HEP seminar, 25 April 2007, the power point presentation available from https://gilkalai.wordpress.com/2009/03/06/noise-sensitivity-lecture-and-tales/, (accessed August 9th, 2022)

  9. Kunita, H., Watanabe, S.: On square-integrable martingales. Nagoya Math. J. 30, 209–245 (1967)

    Article  MathSciNet  Google Scholar 

  10. Le Jan, Y., Raimond, O.: Flows, coalescence and noise. Ann. Probab. 32(2), 1247–1315 (2004). https://doi.org/10.1214/009117904000000207

    Article  MathSciNet  Google Scholar 

  11. Le Jan, Y., Raimond, O.: Sticky flows on the circle and their noises. Probab. Theory Relat. Fields 129(1), 63–82 (2004). https://doi.org/10.1007/s00440-003-0324-9

    Article  MathSciNet  Google Scholar 

  12. Lee, J., Bahri, Y., Novak, R., Schoenholz, S., Pennington, J., Sohldickstein, J.: Deep neural networks as gaussian processes. Published as a conference paper at In International Conference on Learning Representations (ICLR) (2018). (also arXiv:1711.00165v3)

  13. Neal, R. M.: Priors for infinite networks. In: Bayesian Learning for Neural Networks, Lecture Notes in Statistics, 1st ed., Vol. 118. Springer, New York, pp 29–53 (1996)

  14. Ogawa, S.: Quelques propriétés de l’intégrale stochastique du type noncausal. Jpn. J. Appl. Math. 1, 405–416 (1984)

    Article  MathSciNet  Google Scholar 

  15. Ogawa, S.: The stochastic integral of noncausal type as an extension of the symmetric integrals. Jpn. J. Appl. Math. 2, 229–240 (1985). https://doi.org/10.1007/BF03167046

    Article  MathSciNet  Google Scholar 

  16. Salvatier, J., Wiecki, T.V., Fonnesbeck, C.: Probabilistic programming in Python using PyMC3. (2016). https://doi.org/10.7717/peerj-cs.55

  17. Schramm, O., Smirnov, S., Garban, C.: On the scaling limits of planar percolation. Ann. Probab. 39(5), 1768–1814 (2011). https://doi.org/10.1214/11-AOP659

    Article  MathSciNet  Google Scholar 

  18. Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50(12), 1261–1286 (1997)

    Article  MathSciNet  Google Scholar 

  19. Tsirelson, B., Vershik, A.M.: Examples of nonlinear continuous tensor product of measure spaces and non-Fock factorizations. Rev. Math. Phys. 10(1), 81–145 (1998). https://doi.org/10.1142/S0129055X98000045

    Article  MathSciNet  Google Scholar 

  20. Tsirelson, B.: Unitary Brownian motions are linearizable. arXiv preprint (1998). arXiv:math.PR/9806112 (also MSRI Preprint No. 1998-027)

  21. Tsirelson, B.: Scaling limit, noise, stability. In: Picard, J. (Ed.), Lectures on probability theory and statistics, Lecture Notes in Math., 1840, Springer, Berlin, pp. 1–106 (2004). arXiv:math/0301237. [Note: the numbers of sections, propositions, lemmas and theorems refer to the arXiv version.]

  22. Tsirelson, B.: Nonclassical stochastic flows and continuous products. Probab. Surv. 1, 173–298 (2004)

    Article  MathSciNet  Google Scholar 

  23. Warren, J., Watanabe, S.: On Spectra of Noises Associated with Harris flows. Stochastic analysis and related topics in Kyoto, Adv. Stud. Pure Math. 41,: Math, pp. 351–373. Soc. Japan, Tokyo (2004)

  24. Watanabe, S.: Analysis of wiener functionals (Malliavin Calculus) and its applications to heat kernels. Ann. Probab. 30, 1–39 (1987). https://doi.org/10.1214/aop/1176992255

    Article  MathSciNet  Google Scholar 

  25. Watanabe, S.: A simple example of black noise. Bull. Sci. Math. 125(6–7), 605–622 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere appreciations to Professor Arturo Kohatsu-Higa for his valuable comments.

Author information

Authors and Affiliations

  1. Fukuoka University, 8-19-1 Nanakuma, Jônan-ku, Fukuoka, 814-0180, Japan

    T. Amaba

  2. Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama, 700-0005, Japan

    T. Aoyama

  3. Fukuoka University Ohori Senior High School, 1-12-1 Ropponmatsu, Chuô-ku, Fukuoka, 810-0044, Japan

    S. Araki

  4. Aomori University, 2-3-1 Kôbata, Aomori, Aomori, 030-0943, Japan

    S. Eguchi

Authors
  1. T. Amaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Aoyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Araki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Eguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Amaba.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T. Amaba: This work was partially supported by funding from Fukuoka University (Grant No. 197102) and by JSPS KAKENHI Grant Number 22K03345.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amaba, T., Aoyama, T., Araki, S. et al. Simulation example of a black noise. Japan J. Indust. Appl. Math. 41, 191–210 (2024). https://doi.org/10.1007/s13160-023-00594-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-023-00594-7

Keywords

Mathematics Subject Classification

Search

Navigation