Smoothing Properties of Fractional Ornstein-Uhlenbeck Semigroups and Null-Controllability

TL;DR

This paper investigates fractional hypoelliptic Ornstein-Uhlenbeck operators, demonstrating their smoothing effects, and applies these results to establish null-controllability and subelliptic estimates for related parabolic equations.

Contribution

It proves Gevrey regularizing effects for fractional Ornstein-Uhlenbeck semigroups and derives null-controllability and subelliptic estimates from these properties.

Findings

Semigroups exhibit Gevrey regularity.

Null-controllability holds from thick control sets.

Global $L^2$ subelliptic estimates are established.

Abstract

We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on $L^2(\mathbb{R}^n)$ satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global $L^2$ subelliptic estimates for the these operators.