Large Deviations And Related Problems For Non-ergodic Markov Processes

The theory of Large Deviation Principles (LDP) of Markov processes made aprominent contribution in the development of Markov Theories, and has been gainingfruitful results with wide applications. However, most results obtained so far are basedon certain kinds of strong ergodic assumptions, which greatly limit their applications.Therefore, it is an important topic in the LDP theory to relax the engodic assump-tion and has important meaning both theoretically and practically. This study mainlyfocuses on two types of Markov processes—the absorbing Markov chains and a se-quence of difusion processes with singular coefcients. Neither of them are typicalegodic processes.In this thesis, we first provide some large deviation results for the empirical mea-sure and empirical processes of absorbing Markov chains. Then we study their rela-tionships with other important results of absorbing Markov chains, including the re-lationship with Dirichlet eigenvalue and that with quasi-stationary distribution. Weprovide the variational representations of those eigenvalues and absorbing coefcientsand furthermore build up their relationships between quasi-stationary distribution andlarge deviation.In this thesis, It is proven that before absorbing, the empirical measureand empirical processes satisfy weak LDP. Under the assumption of exponential tight-ness, the full LDP follows. We also provide a characterization for the rate function. Wealso get some results with wide adaptability: the exponential tightness of the marginalmeasures and some uniform exponential tightness respectively imply the LDP upperbound and the uniform LDP upper bound of the empirical processes.For one-dimensional difusion processes with singular coefcients, by provingthe comparison principle for viscosity solutions to the corresponding Hamilton-Jacobi(H-J) equations, we obtain the large deviation results and provide the variational rep-resentation of the rate function, which is similar to the cost function in control theory.In particular, we point out when the H-J equations are singular, the traditional distance function does not apply. We construct a new geometrically invariant distance functionand apply it to the proof of the comparison principle.The main contributions of this thesis are:(1) We relax the assumption of strong ergodic and prove LDP for the absorbingMarkov chains.(2) We provide the variational representation of the principal eigenvalue by the ratefunction before absorbing.(3) We identify the relationship between large deviations of absorbing Markov chainsand the quasi-stationary distribution, Dirichlet principal eigenvalue.(4) We prove LDP holds for a sequence of difusion processes with singular coef-cients by proving comparison principle for the corresponding H-J equations.