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

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Abstract
Lett≥ 0 be a Markov transition probability semigroup on a general spacesatisfying a suitableφ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒestPt(x,A)dtfor almost allxand all suitableA, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of theh-skeleton chains of (Pt). We study the convergence ofeλtPt(x, A) in the λ-positive case, and show that the limitf(X)π(A) (wherefand 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. ||eλtPt(x, ·)-f(x)π(·)||f=O(e−αt) for some α > 0 and almost allxif there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (Pt) to have quasi-stationary distributions.

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