An Extension to Gaussian Semigroup and Some Applications

广义Mehler半群的超收缩性性质

Hypercontractivity type property for generalized Mehler semigroups

研究广义Mehler半群在L^p尺度上的超收缩性，探讨其在混合范数空间中的改进性质及修正的对数Sobolev不等式。

On Gaussian Brunn-Minkowski inequalities

In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines.

高斯量子马尔可夫半群的不可约性与正则化性质

Irreducibility and regularisation properties of Gaussian quantum Markov semigroups

该研究分析连续变量量子系统上的高斯量子马尔可夫半群的正则化与不可约性，建立相关代数准则，发现其不可约性强于正则化条件，为量子马尔可夫半群研究提供框架。

Operator-Valued Monotone Convolution Semigroups and an Extension of the Bercovici-Pata Bijection

In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…

有根树上的Ornstein--Uhlenbeck半群

Ornstein--Uhlenbeck semigroup on rooted trees

研究有根度量树上带高斯型测度的Ornstein-Uhlenbeck算子，通过形式方法构造Dirichlet和Neumann实现，证明半群对称、解析、保正，并给出紧性和线性特征值渐近。

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities wit…

Quantum extensions of dynamical systems and of Markov semigroups

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in…

Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups

In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Itô, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor, enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excur…

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isope…

Symmetries of Gaussian measures and operator colligations

Consider an infinite-dimensional linear space equipped with a Gaussian measure and the group $GL(\infty)$ of linear transformations that send the measure to equivalent one. Limit points of $GL(\infty)$ can be regarded as 'spreading' maps (polymorphisms). We show that the closure of $GL(\infty)$ in the semigroup of polymorphisms contains a certain semigroup of operator colligations and write explicit formulas for action of operator colligations by polymorphisms of the space with Gaussian measure.

Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus $a \mapsto a(Q,P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $ L = -Δ+ x\cdot \nabla$, and give a simple criterion for restricted $L^p$-$L^q$ boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of $L$. It allows us to recover, unify, and extend, old and new results concerning the boundedness of $\exp(-zL)$ as an operator from $L^p(\mathbb{R}^d,γ_α)$ to $L^q(\mathbb{R}^d…

实矩阵半群的一个刻画

A characterization of real matrix semigroups

本文在不假设连续性的条件下刻画了满足温和有界性假设的实矩阵半群，借助准素分解与若尔当-谢瓦莱分解描述连续解及由柯西方程非可测解产生的解，动机来自多维自相似高斯马尔可夫过程的刻画。

Asymptotics of smoothed Wasserstein distances

We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$ under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space---where the heat semigroup corresponds to smoothing the measures by Gaussian convolution---the situation is more subtle. We prove precise asymptotics for the $2$-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is alwa…

Noncommutative extensions of the Fourier transform and its logarithm

We introduce and study noncommutative extensions of the Fourier transform and its logarithm to the algebra of functions on the free semigroup FS(2) on two generators with the convolution multiplication. These extensions are new types of moment and cumulant generating functions, respectively, the latter corresponding to the cumulants which are additive under the so-called filtered convolution on the free *-algebra on two generators. This algebra plays the role of a ``noncommutative plane'' built on the ``classical real line'' and the ``boolean real line''. The restrictions of the cumulant gener…

Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup

Let $γ_n$ be the standard Gaussian measure on $\mathbb R^n$ and let $(Q_t)$ be the Ornstein--Ulhenbeck semigroup. Eldan and Lee recently established that for every non--negative function $f$ of integral $1$ and any time $t$ the following tail inequality holds true: \[ γ_n ( \{ Q_t f > r \} ) \leq C_t \, \frac{ (\log \log r)^4 }{r \sqrt{\log r}} , \quad \forall r>1 \] where $C_t$ is a constant depending on $t$ but not on the dimension. The purpose of the present paper is to simplify parts of their argument and to remove the $(\log \log r)^4$ factor.

A unification of the hypercontractivity and its exponential variant of the Ornstein-Uhlenbeck semigroup

Let $γ_{d}$ be the $d$-dimensional standard Gaussian measure and $\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{1}(γ_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge 0}$ is equivalent to the property that \begin{align*} \left\{ \int_{\mathbb{R}^{d}}\exp \left(e^{2t}Q_{t}f\right) dγ_{d} \right\} ^{1/e^{2t}} \le \int_{\mathbb{R}^{d}}e^{f}\,dγ_{d}, \end{align*} which holds for any $f\in L^{1}(γ_{d})$ with $e^{f}\in L^{1}(γ_{d})$ and for any $t\ge 0$. We then derive a family of inequalities that unifies this exponential variant and the original hypercontractivity…