Kolmogorov equations on spaces of measures associated to nonlinear filtering processes

We introduce and study some backward Kolmogorov equations associated to filtering problems. In the stochastic filtering framework, SDEs for measure-valued processes arise naturally (Zakai and Kushner- Stratonovich equation). The associated Kolmogorov equations have been intensively studies, assuming that the measure-valued processes admit a density and then by exploiting stochastic calculus in Hilbert spaces.Our approach differs from this since we do not assume the existence of a density and we work directly in the context of measures. We first formulate two Kolmogorov equations on spaces of measures, and then we prove existence and uniqueness of classical solutions.