Multiscale functional inequalities in probability: Constructive approach
Multiscale functional inequalities in probability: Constructive approach
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概率中的多尺度函数不等式：建设性方法
DOI：
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发表时间：
2017
期刊：
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通讯作者：
A. Gloria
中科院分区：
文献类型：
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作者：
Mitia Duerinckx;A. Gloria
考虑环境空间$mathbb R^d$上的遍历平稳随机场$A$。为了建立非线性函数Z（A）$的集中性质，标准的方法是求助于概率空间中的函数不等式，如Poincare不等式或对数Sobolev不等式。然而，这些不等式只适用于一类有限的定律（乘积测度、具有可积协方差的高斯测度或具有良好表现的哈密顿量的更一般的吉布斯测度）。在这方面的贡献，我们介绍了这些不等式的变种，我们称之为多尺度函数不等式，这仍然意味着精细的浓度属性，我们开发了一个建设性的方法，这样的不平等。我们认为，可以被视为产品结构的转换，其中的问题是减少到设计近似的链式规则的非线性随机变化的变量的随机场。这种方法使我们能够涵盖应用科学中的异质材料建模中产生的随机场的大多数例子，包括具有任意协方差函数的高斯场，具有（无界）随机半径的泊松随机内含物，随机停车和Matern型过程，以及泊松随机镶嵌。所获得的多尺度功能的不平等，我们主要是在这里开发，鉴于其应用的浓度和定量随机均匀化，是独立的利益。
Consider an ergodic stationary random field $A$ on the ambient space $mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincare or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matern-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.