Geodesic
Distribution density of commutant of random rotations of three-dimensional Euclidean space
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 327-347 
F. M. Malyshev. Distribution density of commutant of random rotations of three-dimensional Euclidean space. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 327-347. http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a5/
@article{TVP_2016_61_2_a5,
     author = {F. M. Malyshev},
     title = {Distribution density of commutant of random rotations of three-dimensional {Euclidean} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {327--347},
     year = {2016},
     volume = {61},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The basic measure $\mu $ is defined on the group $ SO (3) $ of rotations of three-dimensional Euclidean space. It responds to the product of uniform distributions on the sets of axes of rotations and angles of rotations. We consider three distribution densities with respect to $\mu$: $\rho_0 $ is a density of left- and right-invariant measure (Haar measure); $ \rho_1 $ is a density of distribution of rotations $ \Lambda^k$, $k \ge 2 $, where $ \Lambda $ is a random rotation with density $ \rho_0 $; and $ \rho_2 $ is a distribution density of the $ \Lambda_1^{- 1} \Lambda_2^{- 1} \Lambda_1 \Lambda_2 $ commutant, where $ \Lambda_1 $, $ \Lambda_2 $ are random independent rotations with the distribution density $ \rho_0 $. It is shown that $ \rho_2 \equiv \sqrt{\rho_0 \rho_1} \frac{\pi \sqrt {2}} {4} $ and the measure $ \mu_1 $ with density $ \rho_1 $ is proportional to the basic measure $ \mu $.

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