  This paper is concerned with the problem of nonlinear filter stability of
ergodic Markov processes. The main contribution is the conditional Poincar\'e
inequality (PI) which is shown to yield filter stability. The proof is based
upon a recently discovered duality result whereby the nonlinear filtering
problem is cast as a stochastic optimal control problem for a backward
stochastic differential equation (BSDE). A comparison is made between the
stochastic stability of a Markov process and the filter stability. The former
is based upon assuming the standard form of PI whereas the latter relies on the
conditional PI introduced in this paper.
