On Steiner Symmetrizations for First Exit Time Distributions

Let $A_t$ be an $\alpha$-stable symmetric process, $0<\alpha\leq 2$, on $\mathbb{R}^d$ and $D\subset \mathbb{R}^d$ be a bounded domain. This paper presents a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B$ and satisfying the $\varepsilon$-cone condition converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from $D_m$ converges to the distribution of the first exit time of Brownian motion from $D'$. These results will then be used to establish inequalities involving distributions of first exit times of $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.