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ºtÁ¿¥DÃD¡GAnalytic Approach for Some Problems in Probability Theory

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Markov processes is a special kind of stochastic processes that are widely used in application. Markov processes is one of the main topics that are extensively studied in modern probability theory. It is interesting to note that the theory of Markov processes is also one of few topics in probability theory which has close connection with other branches of mathematics. For example, diffusion processes are Markov processes that relate to the second order partial differential equations. The diffusion processes is also used in the study of problems from geometry. Such connections provide good sources for many interesting research problems. In this talk, I will mention some examples to illustrate the use of analytic approach to study some problems from Markov processes. From this, a generalized eigenvalue problem is considered that relates to the large time behavior of a Markov process. Operator theory is used to give the exponential convergence of the Markov process to the equilibrium. We use examples to show some interesting developments. This include continuous time finite state Markov chains (the case of finite state space), the continuous time simple random walks on

lattice ( the case of discrete infinite state space), the diffusion processes ( the case of continuous state space).