Log-concavity of solutions of parabolic equations related to the Ornstein-Uhlenbeck operator and applications

TL;DR

This work proves that the parabolic Ornstein–Uhlenbeck kernel in a bounded convex domain is log-concave when weighted by the Gaussian measure, and uses this to show that the OU flow preserves log-concavity of the initial data. The authors derive a kernel-based, Prékopa–driven approach to obtain a Brunn–Minkowski type inequality for the first eigenvalue and to re-establish the log-concavity of the first eigenfunction. By connecting the OU operator to a Schrödinger/harmonic-oscillator framework, they unify parabolic kernel methods with spectral theory and provide kernel representations for the Dirichlet OU semigroup via the Trotter product formula. The results extend classical kernel-based log-concavity ideas (Brascamp–Lieb) to the Gaussian setting and offer tools for convex-domain spectral inequalities in Gauss space.

Abstract

In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As an application, we give another proof of a Brunn-Minkowski type inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and of the log-concavity of the related first eigenfunction (both results have been proved in [9], by different methods).

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Theorem 1.2 in CFLS24
• Theorem 1.4: Theorem 1.7 in CFLS24
• Remark 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Remark 3.4
• Proposition 4.1