On the convex infimum convolution inequality with optimal cost function

Rogozin's convolution inequality for locally compact groups

General extensions of an inequality due to Rogozin, concerning the essential supremum of a convolution of probability density functions on the real line, are obtained. While a weak version of the inequality is proved in the very general context of Polish $σ$-compact groups, particular attention is paid to the group \(\mathbb{R}^d\), where the result can combined with rearrangement inequalities for certain linear images for a strong generalization. As a consequence, we obtain a unification and sharpening of both the \(\infty\)-Renyi entropy power inequality for sums of independent random vector…

On the convolution inequality $f \geq f\star f$

We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are non-negative, which is not the case for the same inequality in $L^p$ for any $1 < p \leqslant 2$, for which the convolution is defined. We also show that all integrable solutions $f$ satisfy $\int f(x){\rm d}x \leqslant \tfrac12$. Moreover, if $\int f(x){\rm d}x = \tfrac12$, then $f$ must decay fairly slowly: $\int |x| f(x){\rm d}x = \infty$, and this is sharp since for all $r< 1$,…

The geometry of Euclidean convolution inequalities and entropy

The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.

MTP2分布函数卷积的一个概率不等式

A Probability Inequality for Convolutions of MTP2-Distribution Functions

本文针对MTP2光滑分布函数的n重卷积证明了一个比高斯相关型不等式更强的概率不等式，并给出由Wishart矩阵对角线导出的多元卡方分布作为重要例子。

Weak Poincaré Inequality for Convolution Probability Measures

By using Lyapunov conditions, weak Poincaré inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.

A Maximal Inequality for $p$th Power of Stochastic Convolution Integrals

An inequality for the $p$th power of the norm of a stochastic convolution integral in a Hilbert space is proved. The inequality is stronger than analogues inequalities in the Literature in the sense that it is pathwise and not in expectation. An application of this inequality is provided for the semilinear stochastic evolution equations with Lévy noise and monotone nonlinear drift. The existence and uniqueness of the mild solutions in $L^p$ for these equations is proved and a sufficient condition for exponential asymptotic stability of the solutions is derived.

A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation …

Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence

The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincar{é} inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.

Tail asymptotics for exponential functionals of Levy processes: the convolution equivalent case

We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Levy measure of the underlying Levy process. The method of proof relies on fluctuation theory of Levy processes and an explicit path-wise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish rather general estimates of the measure of the ex…

An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner's type Poincaré inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $μ\starν$, where $μ$ is the reference Gaussian measure and $ν$ is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and in…

A multiplicative inequality for concentration functions of $n$-fold convolutions

We estimate the concentration functions of $n$-fold convolutions of one-dimensional probability measures. The main result is a supplement to the results of Götze and Zaitsev (1998). We show that the estimation of concentration functions at arguments of bounded size can be reduced to the estimation of these functions at arguments of size $O(\sqrt n)$ which is easier.

Hamilton-Jacobi equations on graph and applications

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.

An inequality for the distance between densities of free convolutions

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $μ_i$ and $ν_i, i=1,2$, are close to each other in terms of the Lévy metric and if the free convolution $μ_1\boxplusμ_2$ is sufficiently smooth, then $ν_1\boxplusν_2$ is absolutely continuous, and the densities of measures $ν_1\boxplusν_2$ and $μ_1\boxplusμ_2$ are close to each other. In particular, convergence in distribution $μ_1^{(n)}\rightarrow μ_1,$ $μ_2^{(n)}\rightarrowμ_2$ implies that the density of $μ_1^{(n)}\boxplusμ_2^{(n)}$ is defined for al…

Bounds on the Poincaré constant for convolution measures

We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the Poincaré constant is non-increasing along the central limit theorem. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson (2004). As a bypr…

Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)ξ(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z…

卷积上对数Sobolev常数的次可加性

Subadditivity of the log-Sobolev constant on convolutions

本文证明了卷积测度对数Sobolev常数的一般次可加性不等式，并由此推出该常数在中心极限定理的标准化卷积序列上具有单调性。

受限几何不等式的函数提升

Functional Liftings of Restricted Geometric Inequalities

本文提出广义上卷积方法，将受限几何不等式提升为函数不等式，并在 Gaussian、对数情形及幂零 Lie 群中给出应用。

卷积下对数超模ularity的稳定性

Four Function Theorems Through Optimal Transport and Entropic Analogs

本文研究了对数超模ularity函数在卷积下的稳定性，并验证了相关猜想，同时提供了两个定理的传输证明及新的泛化结果。

A concentration inequality and a local law for the sum of two random matrices

Let H=A+UBU* where A and B are two N-by-N Hermitian matrices and U is a Haar-distributed random unitary matrix, and let μ_H, μ_A, and μ_B be empirical measures of eigenvalues of matrices H, A, and B, respectively. Then, it is known (see, for example, Pastur-Vasilchuk, CMP, 2000, v.214, pp.249-286) that for large N, measure μ_H is close to the free convolution of measures μ_A and μ_B, where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of μ_H from its expectation have been studied by Chatterjee in in JFA, 200…

Functional Inequalities for Convolution Probability Measures

Let $μ$ and $ν$ be two probability measures on $\R^d$, where $μ(\d x)= \e^{-V(x)}\d x$ for some $V\in C^1(\R^d)$. Explicit sufficient conditions on $V$ and $ν$ are presented such that $μ*ν$ satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, the recent results on the log-Sobolev inequality derived in \cite{Z} for convolutions of the Gaussian measure and compactly supported probability measures are improved and extended.

一个卷积不等式，为Zolotarev-接近正态的被加项导出更精确的Berry-Esseen定理

A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal

通过弱范数距离替代三阶绝对矩，改进Berry-Esseen界，简化Ulyanov和Senatov的结果，并基于卷积不等式给出更精确的正态逼近定理。

带有限多个输出函数的Borell–Brascamp–Lieb不等式

Borell--Brascamp--Lieb inequality with finitely many output functions

本文将经典Borell–Brascamp–Lieb不等式扩展为每个输入函数对应不同输出函数的形式，建立了含加权幂平均、幂变换的新不等式，其特例可还原经典不等式或对应多输出Prékopa–Leindler不等式的特例。

秩-2 永久式与有限自由卷积的不等式

Inequalities for rank-two permanents and finite free convolutions

该研究强化了Bang的矩阵永久式不等式，将其适用范围扩展至秩不超过2的实矩阵，并利用该强化结果推导得出多项式有限自由卷积的新不等式。

Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

We present a new, very short proof of a conjecture by I. Raşa, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. Wąsowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Raşa inequality. We give also the corresponding inequalities for other distributions. Our methods all…

Quadratic Transportation Cost Inequalities Under Uniform Distance For Stochastic Reaction Diffusion Equations Driven by Multiplicative Space-Time White Noise

In this paper, we established a quadratic transportation cost inequality for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise based on a new inequality we proved for the moments (under the uniform norm) of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

Discrete/Continuous Elliptic Harnack Inequality and Kernel Estimates for Functions of the Laplacian on a Graph

This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments for the elliptic case fail in this "mixed" setting, we adapt ideas introduced by Thierry Delmotte for the discrete parabolic case. We then present some useful applications of this inequality, namely, a kernel estimate for functions of the Laplacian on a graph. This application in turn provides sharp estimates for certain Markov kernels on graphs. We then clos…

Convolution of beta prime distribution

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for ${}_3F_2(1)$. Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán's inequality for…

量子Ruzsa分歧用于量化魔法

Quantum Ruzsa Divergence to Quantify Magic

本文研究量子熵在量子卷积下的行为及其在量化量子态魔法方面的应用，提出量子Ruzsa分歧等新方法，扩展了经典逆和集理论到量子领域。