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Bifractional Brownian Motions on Metric Spaces

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Abstract

Fractional and bifractional Brownian motions can be defined on a metric space if the associated metric or distance function is conditionally negative definite (or of negative type). This paper introduces several forms of scalar or vector bifractional Brownian motions on various metric spaces and presents their properties. A metric space of particular interest is the arccos-quasi-quadratic metric space over a subset of \(\mathbb {R}^{d+1}\) such as an ellipsoidal surface, an ellipsoid, or a simplex, whose metric is the composition of arccosine and quasi-quadratic functions. Such a metric is not only conditionally negative definite but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that it enables us to study (bi, tri, quadri)fractional Brownian motions on various metric spaces in a unified manner. The vector fractional Brownian motion on the arccos-quasi-quadratic metric space enjoys an infinite series expansion in terms of spherical harmonics, and its covariance matrix function admits an ultraspherical polynomial expansion. We establish the property of strong local nondeterminism of fractional and bi(tri, quadri)fractional Brownian motions on the arccos-quasi-quadratic metric space.

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Acknowledgements

The author wishes to thank an associate editor and a referee very much for their helpful suggestions and comments which led to a better presentation and to the improved readability of this work.

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  1. Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, 67260-0033, KS, USA

    Chunsheng Ma

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Ma, C. Bifractional Brownian Motions on Metric Spaces. J Theor Probab 37, 1299–1332 (2024). https://doi.org/10.1007/s10959-023-01284-3

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