Probability Theory and Related Fields最新文献_第4页

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-05-16 DOI: 10.1007/s00440-024-01286-0

Dan Mikulincer, Yair Shenfeld

{"title":"The Brownian transport map","authors":"Dan Mikulincer, Yair Shenfeld","doi":"10.1007/s00440-024-01286-0","DOIUrl":"https://doi.org/10.1007/s00440-024-01286-0","url":null,"abstract":"Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"68 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Inhomogeneous long-range percolation in the weak decay regime 弱衰变体系中的非均质长程渗流

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-05-15 DOI: 10.1007/s00440-024-01281-5

Christian Mönch

引用次数: 0

Minimal distance between random orbits 随机轨道之间的最小距离

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-05-11 DOI: 10.1007/s00440-024-01283-3

Sébastien Gouëzel, Jérôme Rousseau, Manuel Stadlbauer

引用次数: 0

Infinite disorder renormalization fixed point for the continuum random field Ising chain 连续随机场伊辛链的无限无序重正化定点

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-04-29 DOI: 10.1007/s00440-024-01284-2

Orphée Collin, Giambattista Giacomin, Yueyun Hu

{"title":"Infinite disorder renormalization fixed point for the continuum random field Ising chain","authors":"Orphée Collin, Giambattista Giacomin, Yueyun Hu","doi":"10.1007/s00440-024-01284-2","DOIUrl":"https://doi.org/10.1007/s00440-024-01284-2","url":null,"abstract":"We consider the continuum version of the random field Ising model in one dimension: this model arises naturally as weak disorder scaling limit of the original Ising model. Like for the Ising model, a spin configuration is conveniently described as a sequence of spin domains with alternating signs (domain-wall structure). We show that for fixed centered external field and as spin-spin couplings become large, the domain-wall structure scales to a disorder dependent limit that coincides with the infinite disorder fixed point process introduced by D. S. Fisher in the context of zero temperature quantum Ising chains. In particular, our results establish a number of predictions that one can find in Fisher et al. (Phys Rev E 64:41, 2001). The infinite disorder fixed point process for centered external field is equivalently described in terms of the process of suitably selected extrema of a Brownian trajectory introduced and studied by Neveu and Pitman (in: Séminaire de probabilités XXIII. Lecture notes in mathematics, vol 1372, pp 239–247, 1989). This characterization of the infinite disorder fixed point is one of the important ingredients of our analysis.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"44 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140838222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Average-case analysis of the Gaussian elimination with partial pivoting 带部分支点的高斯消元的平均情况分析

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-04-22 DOI: 10.1007/s00440-024-01276-2

Han Huang, Konstantin Tikhomirov

{"title":"Average-case analysis of the Gaussian elimination with partial pivoting","authors":"Han Huang, Konstantin Tikhomirov","doi":"10.1007/s00440-024-01276-2","DOIUrl":"https://doi.org/10.1007/s00440-024-01276-2","url":null,"abstract":"The Gaussian elimination with partial pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a typical square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random (ntimes n) standard Gaussian coefficient matrix A, the growth factor of the Gaussian elimination with partial pivoting is at most polynomially large in n with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve (Ax = b) to m bits of accuracy using GEPP is (m+O(log n)), which improves an earlier estimate (m+O(log ^2 n)) of Sankar, and which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"82 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Private measures, random walks, and synthetic data 私人措施、随机漫步和合成数据

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-04-20 DOI: 10.1007/s00440-024-01279-z

March Boedihardjo, Thomas Strohmer, Roman Vershynin

{"title":"Private measures, random walks, and synthetic data","authors":"March Boedihardjo, Thomas Strohmer, Roman Vershynin","doi":"10.1007/s00440-024-01279-z","DOIUrl":"https://doi.org/10.1007/s00440-024-01279-z","url":null,"abstract":"Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex—but very common—machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data in general compact metric spaces, for any fixed privacy budget (varepsilon ) bounded away from zero. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"11 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140627967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Dirichlet forms on unconstrained Sierpinski carpets 无约束 Sierpinski 地毯上的 Dirichlet 形式

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-04-12 DOI: 10.1007/s00440-024-01280-6

Shiping Cao, Hua Qiu

引用次数: 0

Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise 带有分数拉普拉卡的非线性福克-普朗克方程和带有莱维噪声的麦肯-弗拉索夫 SDEs

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-04-12 DOI: 10.1007/s00440-024-01277-1

Viorel Barbu, Michael Röckner

引用次数: 0

The structure of the local time of Markov processes indexed by Lévy trees 以列维树为索引的马尔可夫过程的局部时间结构