A stochastic partial differential equation with values in a manifold

We investigate a certain stochastic partial differential equation which is defined on the unit interval with periodic boundary condition and takes values in a manifold. Such equation has particularly two different applications. Namely, it determines the evolution law of an interacting constrained system of continuum distributed over the unit circle, while it defines a diffusive motion of loops on a manifold. We establish the existence and uniqueness results and then show the smoothness property of the solutions. Some examples are given in the final section.