Limiting behavior of surface measures on spaces of trajectories (Q1889715)

The author investigates the so called Smolyanov measures, i.e., measures on \(C([0,T],M)\) where \(M\) is a compact Riemannian manifold embedded into a Euclidean space or into another manifold. It is shown that for arbitrary \(T\) those measures exist and the Radon-Nikodym density with respect to the Wiener measure is presented. For the case \(\dim M=1\), i.e. \(M\) is a closed curve, the limit for such measures as \(T\to\infty\) is found as the measure corresponding to a solution of a certain stochastic differential equation.