C. Roberto, B. Zegarlinski
We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality.
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In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.
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…
Hypercontractivity, Nash inequalities, and subordination for classes of nonlinear semigroups
给出了非线性半群$\{T_t\}$的超压缩性的一个适当的概念，并证明了它的生成元$H$的Gagliardo-Nirenberg不等式，只要能量泛函$(u,Hu)$具有次齐性.我们利用这一事实证明，对于由$p$ -Laplacian-型算子生成的半群，超压缩性蕴涵超压缩性。当Bernstein函数为$f(x)=x^α$时，我们引入了从属非线性半群的概念，并给出了相应生成元的显式表达式.它表明，在某些情况下，超收缩仍然成立的从属半群，因此，纳什型不等式成立，以及从属发电机。
For $1<p\leq q$ we show that the Poisson semigroup $e^{-t\sqrt{-Δ}}$ on the $n$-sphere is hypercontractive from $L^{p}$ to $L^{q}$ in dimensions $n \leq 3$ if and only if $e^{-t\sqrt{n}} \leq \sqrt{\frac{p-1}{q-1}}$. We also show that the equivalence fails in large dimensions.
Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups
本文研究了一族量子奥本海姆-乌伦贝克半群的超收缩性，并利用梅克斯纳多项式证明了相关时间最优阶的结果。
Hypercontractivity for semigroups of unital qubit channels
证明了属于自伴半群的量子比特通道的乘积的超压缩性。超压缩界给出了乘积e^{- t_1 H_1} \ot.的充要条件。\ot e^{- t_n H_n}是L^p到L^q的一个压缩，其中L^p是2^n维矩阵代数，赋正规Schatten范数，每个生成元H_j是2维矩阵代数上的一个自伴半正定算子.作为一个特殊的情况下，该结果建立了量子比特去极化通道的乘积的超压缩界。
Hypercontractivity for free products
本文得到了作用在Clifford代数上的Ornstein-Uhlenbeck半群的自由积扩张的最优时间超压缩性界。我们的方法是基于一个中心极限定理的自由积的自旋矩阵代数混合交换/反交换关系。利用Speicher的中心极限定理，我们还可以得到在Fermonic框架和Bosonic框架之间插值的q-变形von Neumann代数的自由积的同样的界.这概括了Nelson、Gross、Carlen/Lieb和Biane的工作。我们的主要应用得到了作用在自由群Fn的群代数上的自由Poisson半群的超压缩性界，且在生成元个数上是一致的.
The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.
Quantum Cheeger Inequalities for KMS-Symmetric Quantum Markov Semigroups
本文为本原KMS对称量子马尔可夫半群建立了量子Cheeger不等式，并证明了其超压缩性和对数Sobolev不等式，且应用于完全修正对数Sobolev不等式。
We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated $\mathbb{L}_p$ norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak…
Space-Time Log-Sobolev Inequality and Hypocoercive Hypercontractivity for Underdamped Langevin Dynamics
针对凸限制势的欠阻尼 Langevin 动力学，通过提出时空对数 Sobolev 不等式，证明了在动力学时间尺度上的超压缩性，并得到了 Rényi 散度的最优低余能衰减率。
We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schrödinger operators based on generators of symmetric jump-paring Lévy processes with Kato-class confining potentials. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps, and the related operators include pseudo-differential operators of interest in mathematical physics. We refine these contractivity properties by the concept of $L^p$-ground state domination and its asymptotic version, a…
Residualities and uniform ergodicities of Markov semigroups
本文利用广义Dobrushin遍历系数研究抽象状态空间上一致P-遍历马尔可夫半群集合的剩余性，并进一步探讨其一致平均遍历性。
Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras
本文研究了有限维C*-代数量子自守群上的热半群，证明了其超收缩性、超收缩性和谱间隙不等式，并获得了Sobolev嵌入性质和Hausdorff-Young不等式的精确性。
Substochastic semigroups and densities of piecewise deterministic Markov processes
给出了由Kato-Voigt扰动定理得到的$L^1$上的子随机半群是随机或强稳定的充要条件.我们展示了这样的半群是如何与分段确定的马尔可夫过程，提供了一个概率解释我们的结果，并将其应用到碎裂方程。
Correlation and Brascamp-Lieb inequalities for Markov semigroups
本文建立在几个最近的作品，Brascamp-Lieb不等式的半群证明提供了各种设置（欧几里得空间，领域和对称群）。我们的目标是双重的。首先，我们提供了一个通用的，统一的，基于马尔可夫生成器的框架，以涵盖各种感兴趣的例子超出以前的调查。其次，我们给出了这些不等式中出现意外指数的组合原因。
Evolutionary semigroups on path spaces
该研究引入路径空间上的演化半群概念以推广转移半群至非马尔可夫随机过程，证明其由移位半群与期望算子复合而成，且马尔可夫过程转移半群可扩展为演化半群，还给出非马尔可夫过程相关的实例。
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.
A Remark on Hypercontractive Semigroups and Operator Ideals
本文回答了Johnson和Schechtman \cite{JS}提出的关于$\{-1,1\}^{\NN}$上的超压缩半群的一个问题.更一般地，我们证明了下面的定理。让$1<p<2$。设$(T(t))_{t>0}$是$L_p$（相对于概率空间）上的全纯半群.假设下面的过度收缩的温和形式：对于某个足够大的数$s>0$，$T(s)$从$L_p$到$L_2$是有界的。则对于任何$t>0$，$T(t)$在由将$L_p$映射到$L_2$的算子形成的子集（由${Γ_2}$表示）的$B(L_p)$（由$\bar{Γ_2}$表示）中的范数闭包中（更不用说这些算子通过希尔伯特空间因子）。
Consider the linear space of functions on the binary hypercube and the linear operator $S_δ$ acting by averaging a function over a Hamming sphere of radius $δn$ around every point. It is shown that this operator has a dimension-independent bound on the norm $L_p \to L_2$ with $p = 1+(1-2δ)^2$. This result evidently parallels a classical estimate of Bonami and Gross for $L_p \to L_q$ norms for the operator of convolution with a Bernoulli noise. The estimate for $S_δ$ is harder to obtain since the latter is neither a part of a semigroup, nor a tensor power. The result is shown by a detailed stud…
Functional inequalities and Hamilton-Jacobi Equations in Geodesic Spaces
研究了在真测地度量空间中共轭指数$p\geq 2$，$q\leq 2$的$p$-Talagrand不等式与$q$-对数Sololev不等式之间的联系.借助于一般的Hamilton-Jacobi半群，我们证明了它们是等价的，并且等价于Hamilton-Jacobi半群的超压缩性。我们的结果推广了洛特和维拉尼的结果。在Heisenberg群的亚黎曼背景下，它们可用于推导$p$ -Talagrand不等式.
The Perron-Frobenius Theorem for Markov Semigroups
设$P^V_t$，$t\ge0$是与$C(X)$上的势$V$和马尔可夫半群$P_t$，$t\ge0$相关联的薛定谔半群.对于所有$t\ge0$，同时建立对应于$P^V_t$的谱半径$e^{λ_0t}$的左特征向量和右特征向量的存在性。这是在半群算子没有紧性假设的情况下导出的。
In this paper we prove a Lie-Trotter product formula for Markov semigroups in spaces of measures. We relate our results to "classical" results for strongly continuous linear semigroups on Banach spaces or Lipschitz semigroups in metric spaces and show that our approach is an extension of existing results. As Markov semigroups on measures are usually neither strongly continuous nor bounded linear operators for the relevant norms, we prove the convergence of the Lie-Trotter product formula assuming that the semigroups are locally equicontinuous and tight. A crucial tool we use in the proof is a …
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…
In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive $C_0$-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a li…
Some entropic inequalities for primitive KMS symmetric quantum Markov semigroups
本文证明原始KMS对称量子马尔可夫半群满足MLSI，构造反例说明非对称或非完全情形不满足，并对图基和费米子系统类建立相应不等式。
Bimodule KMS Symmetric Quantum Markov Semigroups and Gradient Flows
本文研究双模KMS对称量子马尔可夫半群，引入方向矩阵并建立梯度流结构，推导出改进的对数Sobolev不等式和Talagrand不等式。
Asymptotic stability for a class of Markov semigroups
设$U\subset K$是紧致度量空间的开稠密子集，$\{Φ_t\}_{t\ge0}$是$U$上的有界Borel可测函数空间上的Markov半群，具有强Feller性质.假设对于每个$x\in\bdu$，在$x$处存在屏障$h\in C(K)$，使得对于所有$t\ge0$都存在$Φ_t(h)\ge h$。还假设每个实值$g\in C(K)$，对于所有$t\ge0$具有$Φ_t(g)\ge g$，并且在$U$内的一个点处达到其全局最大值是常数。则对于每个$f\in C(K)$，存在一致极限$F=\lim_{t\to\infty}Φ_t(f)$。此外，$F$在$K$上是连续的，与$f$在$\partial{U}$上和$Φ_t(F)=F$在所有$t\ge0$上一致。
The first aim of this paper is to introduce a class of Markov chains on $\mathbb{Z}_+$ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on $\mathbb{R}_+$. As a by-product, we prove that each of these Markov chains, after an appropriate s…
In this note we prove a refined version of the Christensen-Evans theorem for generators of uniformly continuous GNS-symmetric quantum Markov semigroups. We use this result to show the existence of GNS-symmetric extensions of GNS-symmetric quantum Markov semigroups. In particular, this implies that the generators of GNS-symmetric quantum Markov semigroups on finite-dimensional von Neumann algebra can be written in the form specified by Alicki's theorem.
Intertwining Properties for Bimodule Quantum Markov Semigroups
本文在量子傅里叶分析框架下，通过梯度形式和迭代梯度形式的傅里叶乘子研究GNS-和KMS-对称半群的Bakry-Émery估计，并系统探讨双模GNS-和KMS-对称量子马尔可夫半群的交织性质及其与Bakry-Émery估计的比较。
Bimodule Quantum Markov Semigroups
本文研究了双模量子马尔可夫半群，提出双模详细平衡条件和KMS对称性，揭示了量子通道结构，并推导了相关不等式。
We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation. This generalizes a result of Cipriani and Sauvageot for tracially symmetric semigroups. Compared to the tracially symmetric case, the derivations in the general case satisfy a twisted product rule, reflecting the non-triviality of their modular group. This twist is captured by the new concept of Tomita bimodules we introduce. If the quantum Markov semigroup satisfies a certain additional regularity condition, the associated Tomita bimodule can be realized inside the $L^2$ space …
Derivations and KMS-Symmetric Quantum Markov Semigroups
本文证明KMS对称量子马尔可夫半群的$L^2$生成元可表示为Hilbert双模值导子的平方，通过引入新的完全正映射推广了迹对称与GNS对称情形的已有结果。
Dilations of markovian semigroups of measurable Schur multipliers
本文利用概率工具证明了自伴保单位完全正可测Schur乘子半群可被更大von Neumann代数上的Markov *-自同构群膨胀，并推导出生成子的泛函演算有界性及插值结果，回答了相关公开问题。
In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.
Intertwining Curvature Bounds for Graphs and Quantum Markov Semigroups
本文引入图与量子马尔可夫半群的交织曲率下界概念，该概念比Bakry-Émery和熵Ricci曲率更强且计算更简单，并在多个例子中验证了该曲率界，改进了去极化半群的熵曲率界。
Second quantization and compactness of induced Gaussian quantum Markov semigroups
该研究刻画了有限模高斯量子马尔可夫半群诱导的希尔伯特-施密特半群的紧性，证明了相关条件等价，推导了范数公式等结果，还通过实例说明紧性与谱间隙的关系。
Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group alge…
We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative $\mathrm{L}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results implies the boundedness of the McIntosh's $\mathrm{H}^\infty$ fun…
We study quantum Dirichlet forms and the associated symmetric quantum Markov semigroups on noncommutative $L^2$ spaces. It is known from the work of Cipriani and Sauvageot that these semigroups induce a first order differential calculus, and we use this differential calculus to define a noncommutative transport metric on the set of density matrices. This construction generalizes both the $L^2$-Wasserstein distance on a large class of metric spaces as well as the discrete transport distance introduced by Maas, Mielke, and Chow-Huang-Li-Zhou. Assuming a Bakry-Émery-type gradient estimate, we sho…
Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state
本研究刻画了有限模上容许正规不变态的高斯量子马尔可夫半群及其不变态集合，附带刻画了有基态的二次哈密顿量，还分析了半群长时间行为、退相干特性及遍历收敛等性质。
This document presents the contents of three lectures delivered by the author at the Erdős Center School ``Optimal Transport on Quantum Structures'', Septemer 19-23, 2022 in Budapest, Hungary. It presents a fairly self contained account of an active topic of current research, and this account should be accessible to most graduate students, as befits lectures for a school. The main results are known, but there a number of new proofs and some new results.
The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups
本文刻画了KMS-对称量子马尔可夫半群生成元的微分结构，给出了有界与无界量子狄利克雷型用导数表示的完整描述，并揭示了模群在定义域上的强连续性及平方根的增生性。
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