论文标题
布朗尼运动的退出时间和第一个dirichlet特征值
Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian
论文作者
论文摘要
对于$ \ mathbb {r}^d $，$ d \ geq 2 $中的域，我们证明了laplacian的频谱底部和下限的通用上限和下限，用于laplacian to power $ p> 0 $，而在$ p $ p $ p $ the of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of brown of布朗运动的时间。结果表明，对于$ p $的整数值，下限是锋利的，而对于$ p \ geq 1 $，上限在渐近上是渐近的，如$ d \ to \ to \ infty $。对于所有$ p> 0 $，我们证明在所有坐标轴相对于凸和对称的域类别中存在一个极端域。对于这类域，我们猜测该立方体是极端的。
For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of $p$ and that for $p \geq 1$, the upper bound is asymptotically sharp as $d\to\infty$. For all $p>0$, we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.
