| Abstrak/Abstract |
The possibility of the incompleteness of differentiable vector fields in stochastic differential equations on a manifold related to exotic differential structures is discussed. It is shown that the incompleteness of vector fields due to the chosen differentiable structure (exotica). In this work, the incompleteness of vector fields will be realized from the fact that the isometric stochastic flows in an exotic differential structure are not equivalent to that of (standard) isometric stochastic
flows. Isometric stochastic flows with the same one point motion in -dimensional standard spheres ܵS7 and 7-dimensional
Gromoll-Meyer exotic spheres are studied, these manifolds are homeomorphic but not diffeomorphic. Therefore, the
incompleteness of vector fields will have an impact on inequivalence between the isometric stochastic flows on standard
sphere ܵS7 and on Gromoll-Meyer sphere sigma GM7 in which the isometric stochastic flows is the solution of the stochastic differential equation itself. In this work, isometric stochastic flows on ܵare generated by polynomial quadratic
which only includes ࣲx1,x2,x3,x4,x5,x6,x7 which is called as a right invariant differential operator In particular, it will be shown that the dimensions of the space of quadratic polynomials operators which induce zero operators on with ܱܵhave dimensions of and ܵisometric stochastic flows with one point motion is a Brownian motion on is represented by a dimensional cube. |
