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 S 7 s and Gromoll-Meyer exotic sphere Σ 7 GM . In this case, S 7 s and Σ 7 GM are homeomorphic but not diffeomorphic. The standard sphere S 7 s can be constructed as the quotient manifold Sp(2, H)/S 3 with the so-called •-action of S 3 , whereas the Gromoll-Meyer exotic sphere Σ 7 GM as the quotient manifold Sp(2, H)/S 3 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 h : S 7 s → Σ 7 GM . The stochastic flow h * g s,t (ω) can be regarded as the same stochastic flow g s,t (ω) on S 7 , but viewed in Gromoll-Meyer differential structure. The flow g s,t (ω) on S 7s and the corresponding flow h * g s,t (ω) on Σ 7 GM 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 Σ 7 GM , the Riemannian metric tensor [ f −1 ] * G on Σ 7 GM , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere S s 7 , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow h * g s,t (ω) 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.