  We continue the investigation of the spectral theory and exponential
asymptotics of Markov processes, following Kontoyiannis and Meyn (2003). We
introduce a new family of nonlinear Lyapunov drift criteria, characterizing
distinct subclasses of geometrically ergodic Markov processes in terms of
inequalities for the nonlinear generator. We concentrate on the class of
"multiplicatively regular" Markov processes, characterized via conditions
similar to (but weaker than) those of Donsker-Varadhan. For any such process
{Phi(t)} with transition kernel P on a general state space, the following are
obtained. 1. SPECTRAL THEORY: For a large class of functionals F, the kernel
Phat(x,dy) = e^{F(x)}P(x,dy) has a discrete spectrum in an appropriately
defined Banach space. There exists a "maximal" solution to the "multiplicative
Poisson equation," defined as the eigenvalue problem for Phat. Regularity
properties are established for \Lambda(F) = \log(\lambda), where \lambda is the
maximal eigenvalue, and for its convex dual. 2. MULTIPLICATIVE MEAN ERGODIC
THEOREM: The normalized mean E_x[\exp(S_t)] of the exponential of the partial
sums {S_t} of the process with respect to any one of the above functionals F,
converges to the maximal eigenfunction. 3. MULTIPLICATIVE REGULARITY: The drift
criterion under which our results are derived is equivalent to the existence of
regeneration times with finite exponential moments for {S_t}. 4. LARGE
DEVIATIONS: The sequence of empirical measures of {Phi(t)} satisfies an LDP in
a topology finer than the \tau-topology. The rate function is \Lambda^* and it
coincides with the Donsker-Varadhan rate function. 5. EXACTR LARGE DEVIATIONS:
The partial sums {S_t} satisfy an exact LD expansion, analogous to that
obtained for independent random variables.
