Eigenvalue gaps for the Cauchy process and a Poincaré inequality
| dc.creator | Banuelos, Rodrigo | |
| dc.creator | Kulczycki, Tadeusz | |
| dc.date | 2004-08-19 | |
| dc.date | 2004-08-26 | |
| dc.date.accessioned | 2026-07-07T05:11:25Z | |
| dc.date.available | 2026-07-07T05:11:25Z | |
| dc.description | A connection between the semigroup of the Cauchy process killed upon exiting a domain $D$ and a mixed boundary value problem for the Laplacian in one dimension higher known as the "mixed Steklov problem," was established in a previous paper of the authors. From this, a variational characterization for the eigenvalues $λ_n$, $n\geq 1$, of the Cauchy process in $D$ was obtained. In this paper we obtain a variational characterization of the difference between $λ_n$ and $λ_1$. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for $λ_* - λ_1$ where $λ_*$ is the eigenvalue corresponding to the "first" antisymmetric eigenfunction for $D$. The proof is based on a variational characterization of $λ_* - λ_1$ and on a weighted Poincaré--type inequality. The Poincaré inequality is valid for all $α$ symmetric stable processes, $0<α\leq 2$, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap $λ_2-λ_1$ in bounded convex domains. | |
| dc.identifier | https://arxiv.org/abs/math/0408267 | |
| dc.identifier | http://arxiv.org/abs/math/0408267 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72232 | |
| dc.subject | Probability | |
| dc.subject | Analysis of PDEs | |
| dc.title | Eigenvalue gaps for the Cauchy process and a Poincaré inequality | |
| dc.type | text |