Abstract: Poincaré (or spectral gap) inequality (PI) is central to the subject of stochastic stability of Markov processes. The PI is the simplest condition which quantifies ergodicity and convergence to stationarity: The Poincaré constant gives the rate of exponential decay. Apart from stochastic stability, the PI has a rich history. It is the fundamental inequality in the study of the elliptic PDEs.
My talk is on the problem of nonlinear filter stability when the hidden Markov process is ergodic. The main contribution is the conditional PI, which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI described in our work, whereas the former relies on the standard form of PI.
This is joint work with Jin Won Kim and Sean Meyn.
