An extension of the "prior density for path" (Onsager-Machlup functional) is defined and shown to exist for Gaussian fields generated by solutions of elliptic PDEs driven by white noise. This functional is then used to define and solve the MAP estimation of such fields observed via nonlinear...

The "prior density for path" (the Onsager-Machlup functional) is defined for solutions of semilinear elliptic type PDEs driven by white noise. The existence of this functional is proved by applying a general theorem of Ramer on the equivalence of measures on Wiener space. As an application, the...