| The aim of the thesis is three folds:a) one is concerning the forward-backward martingale decomposition, a powerful tool introduced by Lyons-Meyer-Zheng on the middle of 80's. We are mainly interested in its extension and its applications in the non-symmetric case;b) large deviations of occupation measure of a non-irreducible Markov process starting from an arbitrary initial measure;c) the representation of additive functionals and local times for jump Markov processes.We firstly extend the forward-backward martingale decomposition of non-symmetric Markov processes given by Liming Wu for real valued additive functionals to Hilbert space values case. This extension allows us to get the functional central limit theorem for empirical measure of a quasi-symmetric Markov process under the minimal condition.Let H be a separable Hilbert space with inner product h and norm denotes an orthonormal basis for H. The H-valued functions will be denoted by the bold letters f, g, .... Let andfi := . We say that f WH-1 ifTheorem 1. Let T = N. There exist three bounded linear mappings where is the future o-field, such that the following forward-backward martingale decomposition holds P-a.s. for every f whereUsing this decomposition we are able to establish the functional central limit theotem.If f WH-1, for any initial measure , as n goes to infinity, the law of the discrete time case, under Pv converges weakly in D([0,1],H) to the law of an H-valued BM (Bt) where the covariance of B1 is given bywhereHere D([0,1],H) is the space of all H-valued cadlag functions on [0,1] equipped with the Skorokhod topology.Secondly, we extend the cross estimate and the tightness result of Lyons-Zheng from the symmetric case to the general stationary situation and obtain a tightness result for laws of Markov processes. The main tool is again the forward-backward martingale decomposition.Let F,G be two open subsets of E such that . We define the energy from F to G as followTheorem 2. Assume that continuous strong Markov processes. We have the following crossing estimateUsing this estimate we can deduce the tightness criteria.Denote by D[0,1] the space of all cadlag functions defined on [0,1] taking values in E. Let E = E U A be the one-point compactification of E. For any subset A C E, A U A is endowed with the relative topology as a subspace of E Let R be the set of all pairs of relatively compact open sets (F, G) with disjoint closures in E. Let (Ln,D(Ln)) be a sequence of generators relative to L2(E,un). We denote by PLn,an the probability measure on D[0,1] induced by the Markov process associated to Ln with an initial probability measure an (not necessarily be equal to un).Theorem 3. Let E be locally compact. Moreover, let an be absolutely continuous with respect to un and supn. If for every pair (F, G) R,then there exists a subsequence n' of n such that which converges weakly on to a law P.Then we generalize the large deviations of Donsker-Varadhan for irreducible Markov processes to general non-irreducible Markov processes.All known results on large deviations of a Markov chain with values in a countable states space are based on the irreducibility. Once the Markov chain is not irreducible as encountered often in biological models, the whole theory of large deviations remains to be built. The main difficulty is that many classical techniques used in the irreducible case can no longer be valid, and the rate function ( a type of entropy ) is to be guessed.Let be a Markov process valued in a Polish space E with transition kernel P(x,dy). Large deviation of occupation measure traditional subject in probability. M. D. Donsker and S. R. S. Varadhan initiated the subject by proving the large deviation principle (in short, LDP) of Px{Ln ) uniformly for initial states x in compact subset under the assumptions of existence, continuity and strict positivity of transition density, and of exponential tightness.Irreducibility assumption, however, is not verified by many interesting ex...
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