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mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces

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  • Daniel Alpay

    (Chapman University)

  • Palle Jorgensen

    (The University of Iowa)

Abstract

Replacing the Lebesgue measure on an interval by a Stieltjes positive non-atomic measure, we study the corresponding counterpart of the Brownian motion. We introduce a new heat equation associated with the measure and make connections with stationary-increments Gaussian processes. We introduce a new transform analysis, and heat equation, associated with the measure, and make connections here too with stationary-increments and stationary Gaussian processes. In the main result of this paper (Theorem 7.2), we use white noise space analysis to derive a new heat equation associated with a (wide class of) stationary-increments Gaussian processes.

Suggested Citation

  • Daniel Alpay & Palle Jorgensen, 2022. "mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2757-2783, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01146-w
    DOI: 10.1007/s10959-021-01146-w
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    References listed on IDEAS

    as
    1. Daniel Alpay & Palle Jorgensen & David Levanony, 2017. "On the Equivalence of Probability Spaces," Journal of Theoretical Probability, Springer, vol. 30(3), pages 813-841, September.
    2. Murad S. Taqqu, 2001. "Bachelier and his times: A conversation with Bernard Bru," Finance and Stochastics, Springer, vol. 5(1), pages 3-32.
    3. Alpay, Daniel & Attia, Haim & Levanony, David, 2010. "On the characteristics of a class of Gaussian processes within the white noise space setting," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1074-1104, July.
    Full references (including those not matched with items on IDEAS)

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