  Consider a stochastic nonlinear system controlled over a possibly noisy
communication channel. An important problem is to characterize the largest
class of channels for which there exist coding and control policies so that the
closed-loop system is stochastically stable. In this paper, we consider the
stability notion of (asymptotic) ergodicity. We prove lower bounds on the
channel capacity necessary to achieve the stability criterion. Under mild
technical assumptions, we obtain that the necessary channel capacity is lower
bounded by the log-determinant of the linearization, double-averaged over the
state and noise space. We prove this bound by introducing a modified version of
invariance entropy and utilizing the almost sure convergence of sample paths
guaranteed by the pointwise ergodic theorem. The fundamental bounds obtained
generalize well-known formulas for linear systems, and are in some cases more
refined than those obtained for nonlinear systems via information-theoretic
methods.
