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ABSTRACT

The coupling method is a crucial tool in the study of Markov processes. We briefly recall the classical coupling method and applications in the study of pure mathematics, then explain the main difficulty to establish Harnack inequality using the coupling method, which leads to the new coupling method called coupling by change of measures.  As applications of the new coupling, the dimension-free Harnack inequality and derivative formulas of Markov semigroups derived. The main idea of the study is illustrated by a simple stochastic differential equation.