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In this article, we define the new concept of local coupling property forMarkov processes and study its relationship with distributional properties ofthe transition probability. In the special case of L\'evy processes we showthat this property is equivalent to the absolute continuity of the transitionprobability and also provide a sufficient condition for it in terms of theL\'evy measure. Our result is stronger than existing results for absolutecontinuity of L\'evy distributions.

Local Coupling Property for Markov Processes with Applications to L\'evy Processes