Small noise asymptotics of nonlinear filters with nonobservable limiting deterministic system

We study the asymptotic behaviour of the Bayesian estimator for a deterministic signal in additive Gaussian white noise, in the case where the set of minima of the Kullback-Leibler information is a submanifold of the parameter space. This problem includes as a special case the study of the asymptotic behaviour of the nonlinear filter, when the state equation is noise-free, and when the limiting deterministic system is nonobservable. We present a practical example where this situation occurs. We give an explicit expression of the limit, as the noise intensity goes to zero, of the posterior probability distribution of the parameter, and we study the rate of convergence.