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Time changed spherical Brownian motions with longitudinal drifts

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  • Ascione, Giacomo
  • Vidotto, Anna

Abstract

We consider a time change of a drifted Brownian motion on the two-dimensional unit sphere. Precisely, we find strong solutions to the related time-nonlocal Kolmogorov equation under suitably regular initial data and we determine the spectral decomposition of its probability density function. Moreover, we study the speed of convergence to the stationary state, proving a non-exponential rate to the equilibrium. Finally, we provide very weak solutions of the same time-nonlocal Kolmogorov equation with general initial data. These results improve the known ones in terms of both the presence of a perturbation and the generality of the initial data.

Suggested Citation

  • Ascione, Giacomo & Vidotto, Anna, 2025. "Time changed spherical Brownian motions with longitudinal drifts," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:spapps:v:181:y:2025:i:c:s0304414924002552
    DOI: 10.1016/j.spa.2024.104547
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    References listed on IDEAS

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    1. Anatoly N. Kochubei & Yuri Kondratiev, 2019. "Growth Equation of the General Fractional Calculus," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
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    4. Chen, Zhen-Qing, 2017. "Time fractional equations and probabilistic representation," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 168-174.
    5. D’Ovidio, Mirko & Leonenko, Nikolai & Orsingher, Enzo, 2016. "Fractional spherical random fields," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 146-156.
    6. Kei Kobayashi, 2011. "Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 24(3), pages 789-820, September.
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