On the Stochastic Flows on $(m+n+1)$-Dimensional Exotic Spheres

Abstract

Stochastic flows of Stratonovich stochastic differential equations on exotic spheres have been studied. The consequences of the choice of exotic differential structure on stochastic processes taking place on the topological space $S^{m+n+1}$ as state space of the processes have been investigated. More precisely, we have investigated the properties of stochastic processes where the state spaces of the stochastic processes under consideration are $({m+n+1})$-dimensional differentiable manifolds which are homeomorphic but not necessarily diffeomorphic to standard ${(m+n+1)}$-dimensional sphere. The differentiable manifolds have been constructed from disjoint union $\mathbb{R}^{m+1}\times S^{n}\sqcup S^m\times \mathbb{R}^{n+1}$ by identifying every pair of its points using a map $u :\mathbb{R}^{m+1}\times S^n\rightarrow S^m\times \mathbb{R}^{n+1}$ which is constructed from a diffeomorphism $h_1\times h_2:S^m\times S^n\rightarrow S^m\times S^n$. The diffeomorphisms $h_1$ and $h_2$, therefore, can be regarded as the carriers of the "exoticism" of the constructed manifolds. For all of the above purposes, homeomorphisms $h$ from the above-constructed manifolds onto the standard sphere explicitly in term of the diffeomorphisms $h_1$ and $h_2$ have been constructed. Using the homeomorphisms $h$ and all their associated maps derived from them and expressed in terms of $h_1$ and $h_2$ as well as their derivatives, we construct the stochastic processes or flows on the above-constructed manifolds corresponding to stochastics processes on the standard sphere $S^{m+n+1}_s$. The stochastic processes yielded from the above construction on the constructed manifolds can be regarded as the same stochastic processes on $S^{m+n+1}_s$ but described in exotic differential structures on $S^{m+n+1}$.