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The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

Author

Listed:
  • Ibrahim Al-Dayel

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia)

  • Sharief Deshmukh

    (Department of Mathematics, King Saud University, Riyadh 11495, Saudi Arabia)

Abstract

We studied the random variable V t = vol S 2 ( g t B ∩ B ) , where B is a disc on the sphere S 2 centered at the north pole and ( g t ) t ≥ 0 is the Brownian motion on the special orthogonal group S O ( 3 ) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of V t for 0 ≤ t ≤ τ , where τ is the first time when V t vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold Γ B seen as the support of the function f ( g ) = vol S 2 ( g B ∩ B ) immersed in S O ( 3 ) . The integral formula depends on the mean curvature of Γ B and the diameter of B .

Suggested Citation

  • Ibrahim Al-Dayel & Sharief Deshmukh, 2023. "The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres," Mathematics, MDPI, vol. 11(8), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1958-:d:1128848
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    References listed on IDEAS

    as
    1. Ming Liao, 1997. "Random Motion of a Rigid Body," Journal of Theoretical Probability, Springer, vol. 10(1), pages 201-211, January.
    2. David R. Brillinger, 1997. "A Particle Migrating Randomly on a Sphere," Journal of Theoretical Probability, Springer, vol. 10(2), pages 429-443, April.
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