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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 $\alpha$-stable process $X$ in bounded ${C}^{1,1}$ domains in ${ℝ}^{n}$, where $0<\alpha <2$ and $n\ge 2$. 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.

##### MSC:
 60J99 Markov processes 60J45 Probabilistic potential theory 60J75 Jump processes 31C99 Generalizations in potential theory