Mathematics and Statistics Vol. 9(3), pp. 290 - 301
DOI: 10.13189/ms.2021.090310
Reprint (PDF) (214Kb)

On the Stochastic Processes on 7-Dimensional Spheres

Nurfa Risha ^1,*, Muhammad Farchani Rosyid ^2
1 Program Studi Pendidikan Fisika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Pendidikan Ganesha, Jalan Udayana No. 10-11, Singaraja 81116, Indonesia
² Departemen Fisika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Gadjah, Yogyakarta, 55281, Indonesia

ABSTRACT

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e., on the standard sphere and Gromoll-Meyer exotic sphere . In this case, and are homeomorphic but not diffeomorphic. The standard sphere can be constructed as the quotient manifold with the so-called -action of S³, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S³. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism . The stochastic flow can be regarded as the same stochastic flow on S⁷, but viewed in Gromoll-Meyer differential structure. The flow on and the corresponding flow on constructed in this paper have the same regularities. There is no difference between the stochastic flow's appearance on S⁷ viewed in standard differential structure and the appearance of the same stochastic flow viewed in the Gromoll-Meyer differential structure. Furthermore, since the inverse mapping h^-1 is differentiable on , the Riemannian metric tensor on , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow and the Fokker-Planck equation associated with the stochastic differential equation have the same regularities provided that the function β is C¹-differentiable. Therefore both differential structures on S⁷ give the same description of the dynamics of the distribution function of the stochastic process understudy on seven spheres.

KEYWORDS
Stochastic Process, Stratonovich Stochastic Differential Equation, Isometric Stochastic Flows, Fokker-Planck Equation, Entropy Rate, Gromoll-Meyer Exotic Sphere

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Nurfa Risha , Muhammad Farchani Rosyid , "On the Stochastic Processes on 7-Dimensional Spheres," Mathematics and Statistics, Vol. 9, No. 3, pp. 290 - 301, 2021. DOI: 10.13189/ms.2021.090310.

(b). APA Format:
Nurfa Risha , Muhammad Farchani Rosyid (2021). On the Stochastic Processes on 7-Dimensional Spheres. Mathematics and Statistics, 9(3), 290 - 301. DOI: 10.13189/ms.2021.090310.