Onsager-Machlup functionals and maximum a posteriori estimation for a class of non-gaussian random fields

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 maximum a posteriori (MAP) estimation problem is considered where the solution of the semilinear equation is observed via a noisy nonlinear sensor. The existence of the optimal estimator and its representation by means of appropriate first-order conditions are derived.