  This paper is concerned with the problem of nonlinear (stochastic) filter
stability of a hidden Markov model (HMM) with white noise observations. A
contribution is the variance decay property which is used to conclude filter
stability. For this purpose, a new notion of the Poincar\'e inequality (PI) is
introduced for the nonlinear filter. PI is related to both the ergodicity of
the Markov process as well as the observability of the HMM. The proofs are
based upon a recently discovered minimum variance duality which is used to
transform the nonlinear filtering problem into a stochastic optimal control
problem for a backward stochastic differential equation (BSDE).
