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Bifractional Brownian Motions on Metric Spaces

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  • Chunsheng Ma

    (Wichita State University)

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 $$\mathbb {R}^{d+1}$$ 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.

Suggested Citation

  • Chunsheng Ma, 2024. "Bifractional Brownian Motions on Metric Spaces," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1299-1332, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01284-3
    DOI: 10.1007/s10959-023-01284-3
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    References listed on IDEAS

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    1. Lei, Pedro & Nualart, David, 2009. "A decomposition of the bifractional Brownian motion and some applications," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 619-624, March.
    2. Tianshi Lu & Chunsheng Ma & Yimin Xiao, 2023. "Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2403-2425, December.
    3. Zuopeng Fu & Yizao Wang, 2020. "Stable Processes with Stationary Increments Parameterized by Metric Spaces," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1737-1754, September.
    4. Volker Krätschmer & Mikhail Urusov, 2023. "A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1454-1486, September.
    5. Talarczyk, Anna, 2020. "Bifractional Brownian motion for H>1 and 2HK≤1," Statistics & Probability Letters, Elsevier, vol. 157(C).
    6. Lan, Xiaohong & Xiao, Yimin, 2018. "Strong local nondeterminism of spherical fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 44-50.
    7. Lan, Xiaohong & Marinucci, Domenico & Xiao, Yimin, 2018. "Strong local nondeterminism and exact modulus of continuity for spherical Gaussian fields," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1294-1315.
    8. Huang, Steel T. & Cambanis, Stamatis, 1979. "Spherically invariant processes: Their nonlinear structure, discrimination, and estimation," Journal of Multivariate Analysis, Elsevier, vol. 9(1), pages 59-83, March.
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