On the Stochastic Processes on $7$-Dimensional Spheres

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. The standard sphere $S_s^7$ can be constructed as the quotient manifold $\mathrm{Sp}(2,\mathbb{H})/S^3$ with the so-called $\bullet$-action of $S^3$, whereas the Gromoll-Meyer exotic sphere ${\Sigma }_{GM}^7$ as the quotient manifold $\mathrm{Sp}(2,\mathbb{H})/S^3$ with respect to the so-called $\star$-action of $S^3$. The Stratonovich stochastic differential equation which describes a continuous-time stochastic process on the standard sphere is constructed and studied. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism $h:S_s^7\to {\Sigma }_{GM}^7$. The corresponding Fokker-Planck equation and entropy rate in the Stratonovich approach is also investigated.

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