O.G. Smolyanov (Moscow State University)
smolian@nw.math.msu.su
Two methods of constructing Brownian motion (and more general diffusion) in (compact) Riemannian manifolds are discussed; both of them use so called surface measures.
The set of trajectories in a Riemannian manifold embedded in a Euclidean space is considered as an infinite-dimensional submanifold of the space of trajectories in this Euclidean space; then the measure on the latter space generated by a diffusion in the Euclidean space induces some surface measures on the submanifold of trajectories. Actually, in this case, there exist two (non-equivalent) natural definitions of surface measures, but they both are absolutely continuous with respect to measures generated by the corresponding diffusions in the Riemannian manifold, and just this fact allows one to use the surface measure to construct diffusions on Riemannian manifolds.
The difference between the methods of constructing such diffusion is implied by the existence of two different definitions of surface measures. The obtained results can be applied in order to develop different versions of Feynman-Kac formulae for the Schrödinger and heat equations on Riemannian manifolds and also to get "Onsager-Macklup functionals" for measures generated by diffusions on manifolds; in the case of vector spaces, the exponents of these functionals can be called the generalised densities of measures and their derivatives coincide with logarithmic derivatives of the measures.
Reference: O.G. Smolyanov, H. v. Weizsaecker, O. Wittich, Diffusions on compact Riemannian manifolds and surface measures, Doklady Math., 61, 2 (2000), 230-237.