[Paper Review] Smoothing Properties of Fractional Ornstein-Uhlenbeck Semigroups and Null-Controllability

Paul Alphonse, Joackim Berniér|arXiv (Cornell University)|Oct 5, 2018

Stability and Controllability of Differential Equations36 references17 citations

TL;DR

This paper establishes Gevrey-type smoothing properties for semigroups generated by fractional Ornstein-Uhlenbeck operators under the Kalman rank condition, proving that solutions gain infinite Gevrey regularity in positive time. This regularization leads to two key results: null-controllability of the associated parabolic equations from thick control subsets and global $L^2$ subelliptic estimates via interpolation theory.

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.

Motivation & Objective

• To analyze the smoothing properties of semigroups generated by fractional Ornstein-Uhlenbeck operators on $L^2(\mathbb{R}^n)$.
• To establish null-controllability of the associated parabolic equations in any positive time from thick control subsets.
• To derive global $L^2$ subelliptic estimates for the operators using interpolation theory.
• To extend known results on hypoellipticity and regularity from the classical ($s=1$) to the fractional ($s>0$) case.
• To unify the analysis of fractional Kolmogorov-type operators and stochastic processes with stable Lévy noise under a common framework.

Proposed method

• The authors analyze the fractional Ornstein-Uhlenbeck operator $\mathcal{P} = \frac{1}{2}\operatorname{Tr}^s(-Q\nabla_x^2) + \langle Bx, \nabla_x \rangle$ on $L^2(\mathbb{R}^n)$, where $s>0$, $Q$ is symmetric positive semidefinite, and $B$ is a matrix.
• They prove that under the Kalman rank condition $\operatorname{Rank}[B \mid Q^{1/2}] = n$, the semigroup $(e^{-t\mathcal{P}})_{t\geq 0}$ exhibits Gevrey regularizing effects.
• The proof relies on spectral analysis and Fourier multiplier techniques to control the decay of the Fourier transform of the semigroup kernel.
• Null-controllability is established by combining the Gevrey smoothing with the theory of thick control sets and duality arguments.
• Subelliptic estimates are derived using interpolation theory between $L^2$ and Gevrey spaces, exploiting the regularizing effect.
• The analysis includes a detailed study of the fractional Kolmogorov operator as a special case, where $\mathcal{P} = v\cdot\nabla_x + (-\Delta_v)^s$.

Experimental results

Research questions

• RQ1 Does the semigroup generated by a fractional Ornstein-Uhlenbeck operator with $s>0$ exhibit Gevrey-type smoothing under the Kalman rank condition?
• RQ2 Can null-controllability be achieved for the parabolic equation associated with $\mathcal{P}$ in any positive time from thick control subsets?
• RQ3 What is the precise nature of the subelliptic regularity of $\mathcal{P}$ in $L^2$-spaces?
• RQ4 How do the smoothing properties of $\mathcal{P}$ compare to those of the classical Ornstein-Uhlenbeck operator ($s=1$)?
• RQ5 What is the role of the Kalman rank condition in ensuring the regularizing and control properties of the semigroup?

Key findings

• The semigroup $e^{-t\mathcal{P}}$ generates Gevrey regularizing effects of order $s$ for all $t>0$ when the Kalman rank condition holds.
• Null-controllability of the parabolic equation $\partial_t u + \mathcal{P}u = 0$ is achieved in any positive time from any thick control subset of $\mathbb{R}^n$, under the Kalman condition.
• Global $L^2$ subelliptic estimates are derived for $\mathcal{P}$ via interpolation between $L^2$ and Gevrey spaces, reflecting the operator's hypoelliptic nature.
• The fractional Kolmogorov operator $v\cdot\nabla_x + (-\Delta_v)^s$ satisfies the Kalman condition and inherits the same smoothing and control properties.
• The equivalence between the Kalman rank condition and the non-degeneracy of the covariance matrix $Q_t$ is established, confirming the hypoellipticity of $\mathcal{P}$.
• The results extend known $L^2$-regularity and control results from the classical case ($s=1$) to the fractional setting ($s>0$), with explicit dependence on the parameter $s$.