  We propose a method for developing the flows of stochastic dynamical systems,
posed as Ito's stochastic differential equations, on a Riemannian manifold
identified through a suitably constructed metric. The framework used for the
stochastic development, viz. an orthonormal frame bundle that relates a vector
on the tangent space of the manifold to its counterpart in the Euclidean space
of the same dimension, is the same as that used for developing a standard
Brownian motion on the manifold. Mainly drawing upon some aspects of the
energetics so as to constrain the flow according to any known or prescribed
conditions, we show how to expediently arrive at a suitable metric, thus
briefly demonstrating the application of the method to a broad range of
problems of general scientific interest. These include simulations of Brownian
dynamics trapped in a potential well, a numerical integration scheme that
reproduces the linear increase in the mean energy of conservative dynamical
systems under additive noise and non-convex optimization. The simplicity of the
method and the sharp contrast in its performance vis-a-vis the correspondent
Euclidean schemes in our numerical work provide a compelling evidence to its
potential.
