The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres
We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of V...
Published in: | Mathematics |
---|---|
Main Authors: | , |
Format: | Text |
Language: | English |
Published: |
Multidisciplinary Digital Publishing Institute
2023
|
Subjects: | |
Online Access: | https://doi.org/10.3390/math11081958 |
id |
ftmdpi:oai:mdpi.com:/2227-7390/11/8/1958/ |
---|---|
record_format |
openpolar |
spelling |
ftmdpi:oai:mdpi.com:/2227-7390/11/8/1958/ 2023-08-20T04:08:37+02:00 The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres Ibrahim Al-Dayel Sharief Deshmukh 2023-04-21 application/pdf https://doi.org/10.3390/math11081958 EN eng Multidisciplinary Digital Publishing Institute Algebra, Geometry and Topology https://dx.doi.org/10.3390/math11081958 https://creativecommons.org/licenses/by/4.0/ Mathematics; Volume 11; Issue 8; Pages: 1958 Brownian motion Lie group heat kernel Riemannian manifold Text 2023 ftmdpi https://doi.org/10.3390/math11081958 2023-08-01T09:46:49Z We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B. Text North Pole MDPI Open Access Publishing North Pole Mathematics 11 8 1958 |
institution |
Open Polar |
collection |
MDPI Open Access Publishing |
op_collection_id |
ftmdpi |
language |
English |
topic |
Brownian motion Lie group heat kernel Riemannian manifold |
spellingShingle |
Brownian motion Lie group heat kernel Riemannian manifold Ibrahim Al-Dayel Sharief Deshmukh The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
topic_facet |
Brownian motion Lie group heat kernel Riemannian manifold |
description |
We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B. |
format |
Text |
author |
Ibrahim Al-Dayel Sharief Deshmukh |
author_facet |
Ibrahim Al-Dayel Sharief Deshmukh |
author_sort |
Ibrahim Al-Dayel |
title |
The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
title_short |
The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
title_full |
The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
title_fullStr |
The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
title_full_unstemmed |
The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres |
title_sort |
heat equation on submanifolds in lie groups and random motions on spheres |
publisher |
Multidisciplinary Digital Publishing Institute |
publishDate |
2023 |
url |
https://doi.org/10.3390/math11081958 |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_source |
Mathematics; Volume 11; Issue 8; Pages: 1958 |
op_relation |
Algebra, Geometry and Topology https://dx.doi.org/10.3390/math11081958 |
op_rights |
https://creativecommons.org/licenses/by/4.0/ |
op_doi |
https://doi.org/10.3390/math11081958 |
container_title |
Mathematics |
container_volume |
11 |
container_issue |
8 |
container_start_page |
1958 |
_version_ |
1774720994977513472 |