Nurfarisha | Searcharxiv

arXiv2019

On the Stochastic Processes on $7$-Dimensional Spheres

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.

math-ph↗

arXiv2019

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

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 $S^{m+n+1}$.

math-ph↗