Variational path-integral representations for the density of a diffusion process

Let p be the density of a diffusion process {x(t)}. A variational representation for p1/2and (p/p)1/2, where p is the density of an invariant measure, is established. More explicitly, these functions are shown to be value functions of stochastic controls problems, where the controlled equation evolves backward in time. The results appear to have considerable potential both from the theoretical and computational point of view. Their extension to reflected diffusions is also considered. This work may be viewed as a rigorous counterpart of the formal Onsager-Machlup theory of nonequilibrium thermodynamics