### Journals Information

**
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 S^{3}, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S^{3}. 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 S^{7}, 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 S^{7} 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 C^{1}-differentiable. Therefore both differential structures on S^{7} 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.