Exponential decay and ergodicity of general Markov processes and their discrete skeletons

Let t ≥ 0 be a Markov transition probability semigroup on a general space satisfying a suitable φ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒ e st P t (x, A) dt for almost all x and all suitable A, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of the h-skeleton chains of (P t ). We study the convergence of e λt P t (x, A) in the λ-positive case, and show that the limit f(X)π(A) (where f and π are the unique λ-invariant function and measure, normalized so that π(f) = 1) is reached at a uniform exponential rate of convergence, i.e. ||e λt P t (x, ·)-f(x)π(·)|| f = O (e −αt ) for some α > 0 and almost all x if there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (P t ) to have quasi-stationary distributions.