Stochastic Equations on Smooth Manifolds

In this chapter, we shall give an invariant construction of stochastic differential equations on smooth manifolds equipped with linear connections and investigate their solutions. We shall pay special attention to the case in which a manifold possesses a vector bundle total space structure and a stochastic equation is compatible with this structure. Finally, we shall construct formal differential extensions of stochastic equations and prove that the solutions of the equations on the considered manifold are smooth with respect to the initial values under some assumptions.