We give asymptotics near the boundary for the distribution of the first exit time of the isotropic alpha-stable Levy process on bounded Lipschitz sets in real euclidean space. These asymptotics bear some relation to the existence of limits in the Yaglom sense of alpha-stable processes. Our approach relies on the uniform integrability of the ratio of Green functions on bounded Lipschitz sets.
We use bounds for the heat remainder to give the first two terms in the small time asymptotic expansion of the trace of the heat kernel of unimodal Levy processes satisfying some weak scaling conditions on bounded Lipschitz domains
