  We study the backward Kolmogorov equation on the space of probability
measures associated to the Kushner-Stratonovich equation of nonlinear
filtering. We prove existence and uniqueness in the viscosity sense and, in
particular, we provide a comparison theorem. In the context of stochastic
filtering it is natural to consider measure-valued processes that satisfy
stochastic differential equations. In the literature, a classical way to
address this problem is by assuming that these measure-valued processes admit a
density. Our approach is different and we work directly with measures. Thus,
the backward Kolmogorov equation we study is a second-order partial
differential equation of parabolic type on the space of probability measures
with compact support. In the literature only few results are available on
viscosity solutions for this kind of problems and in particular the uniqueness
is a very challenging issue. Here we find a viscosity solution and then we
prove that it is unique via comparison theorem.
