Local Dispersive and Strichartz estimates for the Schr\"odinger equation associated to the Ornstein-Uhlenbeck operator

TL;DR

This paper develops localized dispersive and Strichartz estimates for the Schr"odinger equation associated with the Ornstein-Uhlenbeck operator, enabling analysis of nonlinear problems in Gaussian spaces.

Contribution

It introduces the first comprehensive Strichartz framework for the Ornstein-Uhlenbeck Schr"odinger equation, overcoming challenges due to non-translation invariance.

Findings

Established localized $L^1 o L^f$ dispersive estimates using Mehler kernel techniques.

Proved weighted Strichartz estimates in Gaussian $L^p$ spaces via interpolation and $TT^*$-method.

Demonstrated local well-posedness for nonlinear Schr"odinger equations in this setting.

Abstract

In this paper we study the linear and nonlinear Schr\"odinger equations associated with the Ornstein-Uhlenbeck (OU) operator endowed with the Gaussian measure. While classical Strichartz estimates are well-developed for the free Schr\"odinger operator on Euclidean spaces, extending them to non-translation-invariant operators like the OU operator presents significant challenges due to the lack of global dispersive decay. In this work, we overcome these difficulties by deriving localized $L^1 \to L^\infty$ dispersive estimates for the OU Schr\"odinger propagator using Mehler kernel techniques. We then establish a family of weighted Strichartz estimates in Gaussian $L^p$ spaces via interpolation and the abstract $TT^*$-method. As an application, we prove local well-posedness results for the nonlinear Schr\"odinger equation with power-type nonlinearity in both subcritical and critical regimes. Our framework reveals new dispersive phenomena in the context of the OU semigroup and provides the first comprehensive Strichartz theory in this setting.