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

Ibrahim Al-Dayel () and Sharief Deshmukh
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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

Mathematics, 2023, vol. 11, issue 8, 1-15

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 .

Keywords: Brownian motion; Lie group; heat kernel; Riemannian manifold (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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