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The Local Times for Additive Brownian Sheets and the Intersection Local Times for Independent Brownian Sheets

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  • Mingjie Liang

    (School of Information Engineering, Sanming University, Sanming 365004, China)

  • Chenfang Lin

    (School of Education and Music, Sanming University, Sanming 365004, China)

Abstract

A new class of Gaussian random fields is introduced in this article, described as additive Brownian sheets (ABSs), which can be regarded as a type of generalized Brownian sheet encompassing Brownian motions, Brownian sheets, and additive Brownian motions. The existence, joint continuity and the Hölder law of the local times for ABSs are derived under certain conditions, and some results of the intersection local times for two independent Brownian sheets are also given as special cases. Furthermore, the intersection local times for two independent Brownian sheets in a Hida distribution is proved through white noise analysis, and the Wiener chaos expansion of the intersection local times is expressed in terms of S -transform. Additionally, the large deviations for the intersection local time of two independent Brownian sheets are established. The multi-parameter Gaussian random fields have become a core tool for complex system analysis due to their flexible multidimensional modeling capabilities. With the improvement of computational efficiency and interdisciplinary integration, the ABS constructed in this article will unleash greater potential in fields such as metaverse simulation, financial mathematics, climate science, precision medicine, quantum physics, and string theory.

Suggested Citation

  • Mingjie Liang & Chenfang Lin, 2025. "The Local Times for Additive Brownian Sheets and the Intersection Local Times for Independent Brownian Sheets," Mathematics, MDPI, vol. 13(9), pages 1-39, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:9:p:1425-:d:1643272
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    References listed on IDEAS

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    1. Deheuvels, Paul & Peccati, Giovanni & Yor, Marc, 2006. "On quadratic functionals of the Brownian sheet and related processes," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 493-538, March.
    2. Keming Chen & Guillaume Woessner, 2025. "Zeros of the Brownian Sheet," Journal of Theoretical Probability, Springer, vol. 38(2), pages 1-31, June.
    3. Crestaux, Thierry & Le Maıˆtre, Olivier & Martinez, Jean-Marc, 2009. "Polynomial chaos expansion for sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 94(7), pages 1161-1172.
    4. Yaozhong Hu, 1997. "Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities," Journal of Theoretical Probability, Springer, vol. 10(4), pages 835-848, October.
    5. David Nualart & Salvador Ortiz-Latorre, 2007. "Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 20(4), pages 759-767, December.
    6. Florit, Carme & Nualart, David, 1995. "A local criterion for smoothness of densities and application to the supremum of the Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 25-31, January.
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