生灭过程与一维扩散过程的对数sobolev不等式

本文运用加权的Hardy不等式的方法给出了生灭过程与一维扩散过程满足对数Sobolev不等式的显式判别准则。     

Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.     

Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.

     

Mass Transport and Variants of the Logarithmic Sobolev Inequality

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting. The research of A.V. Kolesnikov was supported by RFBR 07-01-00536, DFG Grant 436 RUS 113/343/0 and GFEN 06-01-39003.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.     

Modified Logarithmic Sobolev Inequalities in Discrete Settings

Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the discrete setting of finite Markov chains and graphs. These inequalities turn out to be weaker than the standard log-Sobolev inequality, but stronger than the Poincare’ (spectral gap) inequality. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log-Sobolev constants go to zero with the size of the graph. We also derive a hypercontractivity formulation equivalent to our main modified log-Sobolev inequality. Along the way we survey various recent results that have been obtained in this topic by other researchers.      

扩散过程轨道空间上的对数Sobolev不等式(英语)

设M是连通Riemann流形,Z是M上C′类向量场,L=(△ Z),本文使用Kendall的耦合分析,给出了参考测度为L-扩散过程在t时刻分布的对数Sobolev常数的估计,并由此建立了轨道空间上的对数Sobolev不等式。此外,本文还给出了流形上的对数Sobolev常数的一个上界估计,所获结果,是对文[1],[2]和[3]的相应结果的推广。     

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of Brendle with Euclidean setting.     

Pointwise symmetrization inequalities for Sobolev functions and applications

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.     

Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.     

对称扩散过程的对数Sobolev不等式、指数可积性和熵的指数衰减性(英)

本文讨论了对称扩散过程的指数可积性，熵的指数衰减性与Sobolev不等式之间的等价关系．并给出了对称扩散过程的对数Sobolev不等式成立的一个充分条件．     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on Rⁿ${\mathbb{R}}^n$ with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.     

Optimal Constants in Critical Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

We prove a theorem on the existence of a `second best constant' incritical Sobolev inequalities on compact Riemannian manifolds underthe action of an isometry group. The theorem is then applied toseveral examples initially introduced by different authors.     

Free Wishart Processes

Using a matrix approach, we define free Wishart processes of parameter > 0 and prove a free additivity property and invertibility for > 1. For 1, we show that a free Wishart process is a solution of a SDE of square Bessel process type, driven by a free complex Brownian motion. In the case > 1, we establish existence and uniqueness of a strong solution of such a SDE.     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

Optimal Gaussian Sobolev embeddings

A reduction theorem is established, showing that any Sobolev inequality, involving arbitrary rearrangement-invariant norms with respect to the Gauss measure in Rn, is equivalent to a one-dimensional inequality, for a suitable Hardy-type operator, involving the same norms with respect to the standard Lebesgue measure on the unit interval. This result is exploited to provide a general characterization of optimal range and domain norms in Gaussian Sobolev inequalities. Applications to special instances yield optimal Gaussian Sobolev inequalities in Orlicz and Lorentz(-Zygmund) spaces, point out new phenomena, such as the existence of self-optimal spaces, and provide further insight into classical results.     

The limiting distributions of large heavy Wigner and arbitrary random matrices

A heavy Wigner matrix $X_N$ is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of $\sqrt{N}{X}_N$ tend to infinity with N, as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family ${\mathbf{X}}_N$ of independent heavy Wigner matrices and an independent family ${\mathbf{Y}}_N$ of arbitrary random matrices with a bound condition and converging in ^?-distribution in the sense of free probability. We characterize the possible limiting joint ^?-distributions of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$, giving explicit formulas for joint ^?-moments. We find that they depend on more than the ^?-distribution of ${\mathbf{Y}}_N$ and that in general ${\mathbf{X}}_N$ and ${\mathbf{Y}}_N$ are not asymptotically ^?-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on ${\mathbf{Y}}_N$ and describe the limiting ^?-distribution of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$. We develop this approach for related models and give recurrence relations for the limiting ^?-distribution of heavy Wigner and independent diagonal matrices.