The optimal nonlinear filtering problem for a diffusion process in a
noncompact domain, observed in white noise, is considered. It is assumed that
the process is ergodic, the diffusion coefficient is constant and the
observation is linear. Using known bounds on the conditional density, it is
shown that when the observation noise is sufficiently small, the filter is
exponentially stable, and that the decay rate of the total variation distance
between differently initialized filtering processes tends to infinity as the
noise intensity approaches zero.