In this paper, we continue the investigation on the connection between observability and inverse problems for a class of parabolic equations with unbounded first order coefficients. We prove new logarithmic stability estimates for a class of initial data in the Ornstein-Uhlenbeck equation posed on $L^2\left(\mathbb{R}^N\right)$ with respect to the Lebesgue measure. The proofs combine observability and logarithmic convexity results that include a non-analytic semigroup case. This completes the picture of the recent results obtained for the analytic Ornstein-Uhlenbeck semigroup on $L^2$-space with invariant measure.
This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension, and their generators. We start from the classical Ornstein-Uhlenbeck semigroup in Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for O-U semigroups in Banach spaces.
We prove Schauder type estimates for stationary and evolution equations driven by the classical Ornstein-Uhlenbeck operator in a separable Banach space, endowed with a centered Gaussian measure.
We obtain explicit mean value formulas for the solutions of the diffusion equations associated with the Ornstein-Uhlenbeck and Hermite operators. From these, we derive various useful properties, such as maximum principles, uniqueness theorems and Harnack-type inequalities.
We study semigroups generated by general fractional Ornstein-Uhlenbeck operators acting on $L2(\mathbb R^n)$. We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein-Uhlenbeck operators on the whole space by using interpolation theory.
We investigate the effect of memory terms on the entropy decay of the solutions to equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.
We study a generalized curvature dimension inequality which is suitable for subelliptic Ornstein-Uhlenbeck type operators and deduce convergence to equilibrium in the $L^2$ and entropic sense. The main difficulty is that the operators we consider may not be symmetric. Our results apply in particular to Ornstein-Uhlenbeck operators on two-step Carnot groups.
A symmetry result for the Ornstein-Uhlenbeck operator
1978年E. De Giorgi提出了一个关于椭圆型方程Δu= F '（u）有界解的一维对称性的猜想，这些解在某个方向上是单调的.本文证明了方程Δu - <x，Du> = F '（u）的类似命题，其中Laplacian算子由Ornstein-Uhlenbeck算子代替.我们的定理在对周围空间的维数没有任何限制的情况下成立，这使得我们能够通过极限过程在无限维中得到类似的结果。
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.
We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein--Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein--Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein--Uhlenbeck semigroup on $\mathbb R$.
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ Δw-\frac{1}{2} \langle x,\nabla w\rangle-\fracλ{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $λ\leq 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\geq 4$ and $λ\in \left[\frac{3 n}{2(n-1)},2\right]$.
$L^p$-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators
当系数$c_{ij}$是$R$上的局部有界函数时，证明了$L^p$-$R^{d+1}$上的$\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $的抛物先验估计.我们稍微概括了通常的抛物线假设，并表明，仍然$L^p$ -估计持有的二阶空间导数的$u$。我们还调查的依赖性常数出现在这样的估计从抛物线常数。最后，我们将我们的估计推广到含有非退化Ornstein-Uhlenbeck型算子的抛物型方程。
In this paper, we study an extension problem for the Ornstein-Uhlenbeck operator $L=-Δ+2x\cdot\nabla +n$ and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for $L$ from which Hardy's inequality for fractional powers of $L$ is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new $L^p-L^q$ estimates are obtained for the fractional powers of the Hermite operator.
