  In the present paper we discuss problems concerning evolutions of densities
related to Ito diffusions in the framework of the statistical exponential
manifold. We develop a rigorous approach to the problem, and we particularize
it to the orthogonal projection of the evolution of the density of a diffusion
process onto a finite dimensional exponential manifold. It has been shown by D.
Brigo (1996) that the projected evolution can always be interpreted as the
evolution of the density of a different diffusion process. We give also a
compactness result when the dimension of the exponential family increases, as a
first step towards a convergence result to be investigated in the future. The
infinite dimensional exponential manifold structure introduced by G. Pistone
and C. Sempi is used and some examples are given.
