  We construct a new random probability measure on the sphere and on the unit
interval which in both cases has a Gibbs structure with the relative entropy
functional as Hamiltonian. It satisfies a quasi-invariance formula with respect
to the action of smooth diffeomorphism of the sphere and the interval
respectively. The associated integration by parts formula is used to construct
two classes of diffusion processes on probability measures (on the sphere or
the unit interval) by Dirichlet form methods. The first one is closely related
to Malliavin's Brownian motion on the homeomorphism group. The second one is a
probability valued stochastic perturbation of the heat flow, whose intrinsic
metric is the quadratic Wasserstein distance. It may be regarded as the
canonical diffusion process on the Wasserstein space.
