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# On Steiner Symmetrizations for First Exit Time Distributions

Mar 2023

Given an $\alpha$-stable symmetric process $A_t$ and a bounded domain $D$,the goal of this paper is to show how first exit time distributions of $A_t$from $D$ increase through the use of Steiner symmetrization. It is also shownthat, when a sequence of domains $\{D_m\}$ satisfying the $\varepsilon$-conecondition converges to a domain $D'$ with respect to the Hausdorff metric, thecorresponding sequence of first exit time distributions of Brownian motion from$D_m$ converges to the first exit time distribution of Brownian motion from$D'$. These results will be then applied to establish results on distributionsof first exit times on specific sequences of domains such as triangles andquadrilaterals.

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