  In this paper, we present conditions for the geometric ergodicity and
Wasserstein regularity of non-linear filter processes, which has received
little attention in the literature. While previous studies in the field of
non-linear filtering have mainly focused on unique ergodicity and the weak
Feller property, our work extends these findings in three main directions: (i)
We present conditions on the geometric ergodicity of non-linear filters, (ii)
we obtain further conditions on unique ergodicity for the case where the state
space is compact, and (iii) as a by-product of our analysis, we obtain
Wasserstein continuity of non-linear filters with stronger regularity. We
present a detailed comparison with the literature. Finally, we study the
controlled setup, and as an application of our geometric ergodicity results we
present new conditions on the existence of solutions to the average cost
optimality equation.
