  We describe, in an intrinsic way and using the global chart provided by Ito's
parallel transport, a generalisation of the notion of geodesic (as critical
path of an energy functional) to diffusion processes on Riemannian manifolds.
These stochastic processes are no longer smooth paths but they are still
critical points of a regularised stochastic energy functional. We consider
stochastic geodesics on compact Riemannian manifolds and also on (possibly
infinite dimensional) Lie groups. Finally the question of existence of such
stochastic geodesics is discussed: we show how it can be approached via
forward-backward stochastic differential equations.
