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

Journal of Theoretical Probability volume 35, pages 2757–2783 (2022)Cite this article

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.

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Acknowledgements

Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. It is a pleasure to thank Professor David Levanony and Professor Izchak Lewkowicz for discussions and inspiration. It is also a pleasure to thank the referee for his/her comments which helped improve the presentation of the paper.

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Authors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

    Daniel Alpay

  2. Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Palle Jorgensen

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  1. Daniel Alpay
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  2. Palle Jorgensen
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Correspondence to Daniel Alpay.

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Alpay, D., Jorgensen, P. mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces. J Theor Probab 35, 2757–2783 (2022). https://doi.org/10.1007/s10959-021-01146-w

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  • DOI: https://doi.org/10.1007/s10959-021-01146-w

Keywords

  • Gaussian processes
  • Stationary square-increments
  • Itô calculus
  • Malliavin derivative
  • Stochastic Fourier transform
  • Diffusion
  • Fractal
  • White noise space analysis
  • Reproducing kernels

Mathematics Subject Classification (2020)

  • 60J22
  • 60J70
  • 46E22