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Estimates on Green functions and Poisson kernels for symmetric stable processes. (English) Zbl 0918.60068
Summary: We give lower and upper bound estimates for Green functions and Poisson kernels of a symmetric α-stable process X in bounded C 1,1 domains in n , where 0<α<2 and n2. An exact formula expressing the Poisson kernel of X on an arbitrary bounded domain D satisfying uniform exterior cone condition in terms of the Green function of X in D is derived. As examples of applications of these estimates, we prove that the 3G Theorem holds for X on bounded C 1,1 domains and that the conditional lifetimes for X in a bounded C 1,1 domain are uniformly bounded. A simple proof of the boundary Harnack principle for nonnegative functions which are harmonic in a bounded C 1,1 domain D with respect to the symmetric stable process is also given.

60J99Markov processes
60J45Probabilistic potential theory
60J75Jump processes
31C99Generalizations in potential theory