ERGODICITY CONDITIONS FOR NONLINEAR DISCRETE-TIME STOCHASTIC DYNAMICAL-SYSTEMS WITH MARKOVIAN NOISE

Let {xi(n) ; n greater-than-or-equal-to 0} be an ergodic Markovian noise process with invariant measure pi(xi). Sufficient conditions are given for the ergodicity of the pair Markov process {(x(n), xi(n)) ; n greater-than-or-equal-to 0} arising from the nonlinear stochastic difference equation x(n+1) = f(x(n), xi(n)) on a manifold M and with f being at least a continuous map. An essential condition for ergodicity is the existence of a unique maximal invariant control set C for the associated control system x(n+1) = f(x(n), u(n)), u(n) is-an-element-of supp (pi(xi)), the support of pi(xi). It is shown that under some further hypotheses involving compactness and weak stochastic controllability, the set C x supp (pi(xi)) is Harris recurrent for the pair process {(x(n), xi(n)) ; n greater-than-or-equal-to 0} and that the invariant measure on C x supp (pi(xi)) is unique and finite. Since geometric control theory is used to prove ergodicity, several results concerning control sets of discrete time dynamical systems are also given.