Patrick Cattiaux
$μ$ being a nonnegative measure satisfying some log-Sobolev inequality, we give conditions on F for the measure $ν=e^{-2F} μ$ to also satisfy some log-Sobolev inequality. Explicit examples are studied.
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In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$ converges exponentially to its unique invariant probability measure $μ$ in entropy, $L^2 (μ)$ and $\|\cdot\|_{\mbox{var}}$, respectively.
The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems. Consequently, the Markov semigroup is exponentially convergent to the invariant probability measure in entropy (thus, also in $L^2$), and is compact for large time. Since the log-Sobolev inequality is invalid for the associated Dirichlet form, we introduce a general result on the hypercontractivity using the Harnack inequality with power. The main results are illustrated by concrete examples.
An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated to a class of functional stochastic differential equations. Consequently, the semigroup $P_t$ converges exponentially to its unique invariant probability measure $μ$ in entropy, $L^2(μ)$ and the totally variational norm, and it is compact in $L^2(μ)$ for large $t>0$. This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated.
We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the an…
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.
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.
We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general $F$-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures $μ…
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 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。
Quantum Reverse Hypercontractivity for KMS-Symmetric Quantum Markov Semigroups
本文研究KMS对称量子马尔可夫半群的反向超压缩性，构造了阻碍正一致率的例子，并证明在特定参数区间内所有本原半群满足该性质，且对图情形给出结果。
We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of Talagrand's $L^1$-$L^2$ inequality for increasing and concave mappings, and we build examples showing that such an estimate represents a strict improvement of the classical Poincaré inequality. We complement our finding with several results of independent interest, such as gradient estimates.
Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equations driven by, respectively, non-degenerate and degenerate Gaussian noises. Consequently, these conditions imply that the associated Markov semigroup is $L^2$-compact and exponentially convergent to the stationary distribution in entropy, variance and total variational norm. As the log-Sobolev inequality is invalid under the framework, we apply a criterion presented in the recent paper \cite{Wang14} using Harnack inequality, coupling property and Gaussian c…
In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher's central limit theorem, we may also obtain the same bounds for free products of q-deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields …
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.
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。
By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the $L^p$-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.
In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work [9], in connection with the study of Schrödinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton-Jacobi-Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are al…
An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional $C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family of operators $L_t=Δ_t +Z_t $ where $Δ_t$ denotes the Laplacian with respect to the metric $g_t$. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by $L_t$, and for existence of evolution systems of probability measures as…
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…
Logarithmic Sobolev inequality and hypercontractivity for the Navier-Stokes Fokker-Planck operator
本文证明随机不可压 Navier-Stokes 方程的 Fokker-Planck 生成元满足对数 Sobolev 不等式（最优常数与纯 Ornstein-Uhlenbeck 算子相同）且其半群具有与 Ornstein-Uhlenbeck 半群相同的超压缩率，关键在于对流生成元在 Gibbs 测度下的反对称性。
We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [7],[9],[10],[11], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise represe…
We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove existence of the Fomin derivative of the corresponding transition semigroup $P_t$. The main tool is a new estimate for $P_tDφ$ in terms of $\|φ\|_{L^2(H,ν)}$, where $ν$ is the invariant measure of $P_t$.
We consider the Burgers equation on $H=L^2(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_t$. We prove a new formula for $P_tDφ$ (where $φ:H\to\R$ is bounded and Borel) which depends on $φ$ but not on its derivative. Then we deduce some new consequences for the invariant measure $ν$ of $P_t$ as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.
Essential spectrum for Brox-type diffusion processes
研究多维布罗克斯扩散过程在两种随机介质下马尔可夫半群的本质谱与紧性，通过多种理论和准则分析，给出高斯和列维环境下相关结论，建立样本路径与谱特征联系框架并提出大势指数开放问题。
Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$
针对任意$p \u2265 2$的$L^p$空间中带耗散漂移项和加性不规则噪声的随机反应扩散方程，推导哈拿克不等式，用测度变换耦合方法研究其马尔可夫半群遍历性，证明超线性漂移（负首项）且受Lipschitz扰动时半群有唯一遍历不变测度。
Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups
本文证明了对于均匀概率测度下的循环群Z_n，带词长ψ_n(k)=min(k,n-k)的拉普拉斯算子满足尖锐对数Sobolev不等式，常数2π与n无关（n≥4）。
Talagrand's convolution conjecture up to loglog via perturbed reverse heat
本文证明在布尔超立方体上的热半群下，非负函数的尾界优于马尔可夫不等式，解决了塔拉甘卷积猜想，方法包括反向热过程、耦合构造和时间平滑反集中估计。
Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations
本文研究非线性无穷维退化随机微分方程，用预解式方法构造无穷维Langevin动力学弱解，结合亚耗散性方法与算子本质m-耗散性结果，证明L²指数遍历性，并应用于显式退化扩散方程。
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