  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 $S^7_s$ 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 $\Sigma^7_{GM}$ 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 $h: S^7_s\rightarrow \Sigma^7_{GM}$. The
corresponding Fokker-Planck equation and entropy rate in the Stratonovich
approach is also investigated.
