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

Daniel Alpay; Palle Jorgensen

doi:10.1007/s10959-021-01146-w

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

Daniel Alpay, Palle Jorgensen

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	2757-2783
Number of pages	27
Journal	Journal of Theoretical Probability
Volume	35
Issue number	4
DOIs	https://doi.org/10.1007/s10959-021-01146-w
State	Published - 1 Dec 2022
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Mathematics (all)
Statistics, Probability and Uncertainty

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10.1007/s10959-021-01146-w

Cite this

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

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.",

keywords = "Diffusion, Fractal, Gaussian processes, It{\^o} calculus, Malliavin derivative, Reproducing kernels, Stationary square-increments, Stochastic Fourier transform, White noise space analysis",

author = "Daniel Alpay and Palle Jorgensen",

note = "Funding Information: 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. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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doi = "10.1007/s10959-021-01146-w",

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T1 - mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces

AU - Alpay, Daniel

AU - Jorgensen, Palle

N1 - Funding Information: 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. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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.

AB - 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.

KW - Diffusion

KW - Fractal

KW - Gaussian processes

KW - Itô calculus

KW - Malliavin derivative

KW - Reproducing kernels

KW - Stationary square-increments

KW - Stochastic Fourier transform

KW - White noise space analysis

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U2 - 10.1007/s10959-021-01146-w

DO - 10.1007/s10959-021-01146-w

M3 - Article

AN - SCOPUS:85123463692

VL - 35

SP - 2757

EP - 2783

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 4

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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