  We derive the (d-dimensional) periodic incompressible and viscous
Camassa-Holm equation as well as the Leray-alpha equations via a stochastic
variational principle. We discuss the existence of solution for this equation
in the space H1 using the probabilistic characterisation. The underlying
Lagrangian flows are diffusion processes living in the group of diffeomorphisms
of the torus. We study in detail these diffusions.
