Sharp Fundamental Gap Estimate on Convex Domains in Gaussian Spaces

Abstract

We prove a sharp lower bound for the fundamental gap on convex domains in Gaussian spaces, the difference between the first two eigenvalues of the Ornstein-Uhlenbeck operator with Dirichlet boundary conditions. Our main result establishes that the gap is bounded below by the gap of the corresponding one-dimensional model, confirming the Gaussian analogue of the fundamental gap conjecture. Furthermore, we demonstrate that the normalized gap of the one-dimensional model is monotonically increasing with the diameter and prove the sharpness of our estimate. Beyond the fundamental gap, we also establish improved log-concavity properties for the Dirichlet heat kernel on convex domains in Gaussian spaces. Our work on Gaussian spaces complements the existing results of Andrews and Clutterbuck and Ni for Euclidean domains, as well as the work of Seto, Wang, and Wei for spherical domains.