Stochastic systems in Riemannian manifolds

A formulation of stochastic systems in a Riemannian manifold is given by stochastic differential equations in the tangent bundle of the manifold. Brownian motion is constructed in a compact Riemannian manifold as well as the horizontal lift of this process to the bundle of orthonormal frames. The solution of some stochastic differential equations in the tangent bundle of the manifold is defined by the transformation of the measure for the manifold-valued Brownian motion by a suitable Radon-Nikodym derivative. Real-valued stochastic integrals are defined for this Brownian motion using parallelism along the Brownian paths. A stochastic control problem is formulated and solved for these stochastic systems where a suitable convexity condition is assumed.This research was supported by NSF Grants Nos. GK-32136, ENG-75-06562, and MCS-76-01695.The author wishes to thank D. Gromoll, J. Simons, and J. Thorpe for some helpful conversations on differential geometry.