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Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds 光滑紧凑黎曼流形中内在弗雷谢特手段的中心极限定理
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-06-24 DOI: 10.1007/s00440-024-01291-3
Thomas Hotz, Huiling Le, Andrew T. A. Wood
{"title":"Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds","authors":"Thomas Hotz, Huiling Le, Andrew T. A. Wood","doi":"10.1007/s00440-024-01291-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01291-3","url":null,"abstract":"<p>We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507361","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
Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space 带有乘法空间白噪声的二维非线性薛定谔方程在全空间上的全局好求解性
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-06-22 DOI: 10.1007/s00440-024-01288-y
Arnaud Debussche, Ruoyuan Liu, Nikolay Tzvetkov, Nicola Visciglia
{"title":"Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space","authors":"Arnaud Debussche, Ruoyuan Liu, Nikolay Tzvetkov, Nicola Visciglia","doi":"10.1007/s00440-024-01288-y","DOIUrl":"https://doi.org/10.1007/s00440-024-01288-y","url":null,"abstract":"<p>We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507312","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
Geometry of Gaussian free field sign clusters and random interlacements 高斯自由场符号集群和随机交错的几何形状
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-06-15 DOI: 10.1007/s00440-024-01285-1
Alexander Drewitz, Alexis Prévost, Pierre-François Rodriguez
{"title":"Geometry of Gaussian free field sign clusters and random interlacements","authors":"Alexander Drewitz, Alexis Prévost, Pierre-François Rodriguez","doi":"10.1007/s00440-024-01285-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01285-1","url":null,"abstract":"<p>For a large class of amenable transient weighted graphs <i>G</i>, we prove that the sign clusters of the Gaussian free field on <i>G</i> fall into a regime of <i>strong supercriticality</i>, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices such as <span>({mathbb {Z}}^d)</span>, for <span>(dge 3)</span>, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman (Ann Math 171(3):2039–2087, 2010), and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from Sznitman (Invent Math 187(3):645–706, 2012).</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507313","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
Characterizing models in regularity structures: a quasilinear case 正则结构中的模型特征：准线性案例
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-06-07 DOI: 10.1007/s00440-024-01292-2
Markus Tempelmayr
{"title":"Characterizing models in regularity structures: a quasilinear case","authors":"Markus Tempelmayr","doi":"10.1007/s00440-024-01292-2","DOIUrl":"https://doi.org/10.1007/s00440-024-01292-2","url":null,"abstract":"<p>We give a novel characterization of the centered model in regularity structures which persists for rough drivers even as a mollification fades away. We present our result for a class of quasilinear equations driven by noise, however we believe that the method is robust and applies to a much broader class of subcritical equations. Furthermore, we prove that a convergent sequence of noise ensembles, satisfying uniformly a spectral gap assumption, implies the corresponding convergence of the associated models. Combined with the characterization, this establishes a universality-type result.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550772","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
Benign overfitting and adaptive nonparametric regression 良性过拟合和自适应非参数回归
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-06-06 DOI: 10.1007/s00440-024-01278-0
Julien Chhor, Suzanne Sigalla, Alexandre B. Tsybakov
{"title":"Benign overfitting and adaptive nonparametric regression","authors":"Julien Chhor, Suzanne Sigalla, Alexandre B. Tsybakov","doi":"10.1007/s00440-024-01278-0","DOIUrl":"https://doi.org/10.1007/s00440-024-01278-0","url":null,"abstract":"<p>We study benign overfitting in the setting of nonparametric regression under mean squared risk, and on the scale of Hölder classes. We construct a local polynomial estimator of the regression function that is minimax optimal on a Hölder class with any given smoothness, and that is a continuous function interpolating the set of observations with high probability. The key element of the construction is the use of singular kernels. Moreover, we prove that adaptation to unknown smoothness is compatible with benign overfitting. Namely, we construct a continuous and interpolating local polynomial estimator attaining the minimax optimal rate in <span>(L_2)</span> adaptively to the unknown Hölder smoothness. Our results highlight the fact that interpolation can be fundamentally decoupled from bias-variance tradeoff in the problem of nonparametric regression.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550773","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
Asymptotics of generalized Bessel functions and weight multiplicities via large deviations of radial Dunkl processes 通过径向 Dunkl 过程的大偏差实现广义贝塞尔函数和权重乘数的渐近性
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-05-19 DOI: 10.1007/s00440-024-01282-4
Jiaoyang Huang, Colin McSwiggen
{"title":"Asymptotics of generalized Bessel functions and weight multiplicities via large deviations of radial Dunkl processes","authors":"Jiaoyang Huang, Colin McSwiggen","doi":"10.1007/s00440-024-01282-4","DOIUrl":"https://doi.org/10.1007/s00440-024-01282-4","url":null,"abstract":"<p>This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-<i>N</i> asymptotics of generalized Bessel functions, as well as a large deviations principle for the more general family of radial Heckman–Opdam processes. As an application, we prove a theorem on the asymptotic behavior of weight multiplicities of irreducible representations of compact or complex simple Lie algebras in the limit of large rank. The theorems in this paper generalize several known results describing analogous asymptotics for Dyson Brownian motion, spherical matrix integrals, and Kostka numbers.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146390","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
The Brownian transport map 布朗运动图
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":"<p>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</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"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
{"title":"Inhomogeneous long-range percolation in the weak decay regime","authors":"Christian Mönch","doi":"10.1007/s00440-024-01281-5","DOIUrl":"https://doi.org/10.1007/s00440-024-01281-5","url":null,"abstract":"<p>We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the <i>weak decay regime</i>, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060039","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
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
{"title":"Minimal distance between random orbits","authors":"Sébastien Gouëzel, Jérôme Rousseau, Manuel Stadlbauer","doi":"10.1007/s00440-024-01283-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01283-3","url":null,"abstract":"<p>We study the minimal distance between two orbit segments of length <i>n</i>, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140929907","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
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":"<p>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 (<i>domain-wall structure</i>). 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 <i>infinite disorder fixed point</i> 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 <i>suitably selected</i> 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</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"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
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