The notion of convexity and concavity on Wiener space

非中心伽马和与差的对数凸性与对数凹性

Log-convexity and log-concavity of noncentral gamma sums and differences

研究非中心伽马随机变量和与差的密度函数的对数凸性与对数凹性，给出相关完整分类及准则，还推导出特殊情况下正态、方差-伽马及麦凯I型分布密度的对数凸凹性分类。

Log-concavity and log-convexity of moments of averages of i.i.d. random variables

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

Representation, relaxation and convexity for variational problems in Wiener spaces

We show convexity of solutions to a class of convex variational problems in the Gauss and in the Wiener space. An important tool in the proof is a representation formula for integral functionals in this infinite dimensional setting, that extends analogous results valid in the classical Euclidean framework.

The Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS

In this paper, we prove that for the doubly symmetric binary distribution, the lower increasing envelope and the upper envelope of the minimum-relative-entropy region are respectively convex and concave. We also prove that another function induced the minimum-relative-entropy region is concave. These two envelopes and this function were previously used to characterize the optimal exponents in strong small-set expansion problems and strong Brascamp--Lieb inequalities. The results in this paper, combined with the strong small-set expansion theorem derived by Yu, Anantharam, and Chen (2021), and …

Roles of Log-concavity, log-convexity, and growth order in white noise analysis

In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence $\{\a(n)\}$ first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural …

Optimal transport on the classical Wiener space with different norms

In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1 are considered apart). The second one is to prove 1-convexity (resp. C-convexity) along (constant speed) geodesics of relative entropy in (P2(W);W2), where W is endowed with the infinite norm (resp. with (k,gamma) norm), and W2 is the 2-distance of Wasserstein.