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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian
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by Rodrigo Bañuelos, Phanuel Mariano and Jing Wang
Trans. Amer. Math. Soc. 376 (2023), 5409-5432
DOI: https://doi.org/10.1090/tran/8903
Published electronically: April 3, 2023

Abstract:

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.
References
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Bibliographic Information
  • Rodrigo Bañuelos
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • Email: banuelos@purdue.edu
  • Phanuel Mariano
  • Affiliation: Department of Mathematics, Union College, Schenectady, New York 12308
  • MR Author ID: 1056562
  • Email: marianop@union.edu
  • Jing Wang
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • Email: jingwang@purdue.edu
  • Received by editor(s): February 19, 2022
  • Received by editor(s) in revised form: October 26, 2022
  • Published electronically: April 3, 2023
  • Additional Notes: Research of the first author was supported in part by NSF Grant DMS-1854709
    Research of the second author was supported in part by an AMS-Simons Travel Grant 2019-2023.
    Research of the third author was supported in part by NSF Grant DMS-1855523
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5409-5432
  • MSC (2020): Primary 60J60, 35P15; Secondary 60J45, 58J65, 35J25, 49Q10
  • DOI: https://doi.org/10.1090/tran/8903
  • MathSciNet review: 4630749