Probability Theory and Related Fields最新文献

{"title":"Harnack inequality and one-endedness of UST on reversible random graphs","authors":"Nathanaël Berestycki, Diederik van Engelenburg","doi":"10.1007/s00440-023-01239-z","DOIUrl":"https://doi.org/10.1007/s00440-023-01239-z","url":null,"abstract":"Abstract We prove that for recurrent, reversible graphs, the following conditions are equivalent: (a) existence and uniqueness of the potential kernel, (b) existence and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack inequality, and (d) one-endedness of the wired uniform spanning tree. In particular this gives a proof of the anchored (and in fact also elliptic) Harnack inequality on the UIPT. This also complements and strengthens some results of Benjamini et al. (Ann Probab 29(1):1–65, 2001). Furthermore, we make progress towards a conjecture of Aldous and Lyons by proving that these conditions are fulfilled for strictly subdiffusive recurrent unimodular graphs. Finally, we discuss the behaviour of the random walk conditioned to never return to the origin, which is well defined as a consequence of our results.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":" 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292693","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}

{"title":"Optimal transport methods for combinatorial optimization over two random point sets","authors":"Michael Goldman, Dario Trevisan","doi":"10.1007/s00440-023-01245-1","DOIUrl":"https://doi.org/10.1007/s00440-023-01245-1","url":null,"abstract":"Abstract We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in $$mathbb {R}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> where the edge cost between two points is given by a p th power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and $$d ge 3, 1le p<d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> . Previous results were limited to the range $$p<d/2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"37 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135430948","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}

{"title":"On the lack of Gaussian tail for rough line integrals along fractional Brownian paths","authors":"H. Boedihardjo, X. Geng","doi":"10.1007/s00440-023-01242-4","DOIUrl":"https://doi.org/10.1007/s00440-023-01242-4","url":null,"abstract":"Abstract We show that the tail probability of the rough line integral $$int _{0}^{1}phi (X_{t})dY_{t}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where ( X , Y ) is a 2D fractional Brownian motion with Hurst parameter $$Hin (1/4,1/2)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$phi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> is a $$C_{b}^{infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> -function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $$gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>γ</mml:mi> </mml:math> -Weibull tail with any exponent $$gamma &gt;2H+1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>H</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $$C_{b}^{infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> . This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"23 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136106006","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}

{"title":"Quantitative stability of barycenters in the Wasserstein space","authors":"Guillaume Carlier, Alex Delalande, Quentin Merigot","doi":"10.1007/s00440-023-01241-5","DOIUrl":"https://doi.org/10.1007/s00440-023-01241-5","url":null,"abstract":"Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a Hölder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that allow to quantify the strong convexity of the barycenter functional. Consequences regarding the statistical estimation of Wasserstein barycenters and the convergence of regularized Wasserstein barycenters towards their non-regularized counterparts are explored.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"51 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972405","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}

{"title":"Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters","authors":"Tom Hutchcroft","doi":"10.1007/s00440-023-01240-6","DOIUrl":"https://doi.org/10.1007/s00440-023-01240-6","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136032488","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}

{"title":"On the pitchfork bifurcation for the Chafee–Infante equation with additive noise","authors":"Alex Blumenthal, Maximilian Engel, Alexandra Neamtu","doi":"10.1007/s00440-023-01235-3","DOIUrl":"https://doi.org/10.1007/s00440-023-01235-3","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946884","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}

{"title":"Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise","authors":"Martina Hofmanová, Theresa Lange, Umberto Pappalettera","doi":"10.1007/s00440-023-01233-5","DOIUrl":"https://doi.org/10.1007/s00440-023-01233-5","url":null,"abstract":"Abstract We construct Hölder continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many Hölder continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $$[0,infty )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> directly.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135648195","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}

{"title":"Mesoscopic central limit theorem for non-Hermitian random matrices","authors":"Giorgio Cipolloni, László Erdős, Dominik Schröder","doi":"10.1007/s00440-023-01229-1","DOIUrl":"https://doi.org/10.1007/s00440-023-01229-1","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387477","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}

{"title":"The random walk on upper triangular matrices over $$mathbb {Z}/m mathbb {Z}$$","authors":"Evita Nestoridi, Allan Sly","doi":"10.1007/s00440-023-01228-2","DOIUrl":"https://doi.org/10.1007/s00440-023-01228-2","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135471802","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}

{"title":"Stationary local random countable sets over the Wiener noise","authors":"Matija Vidmar, Jon Warren","doi":"10.1007/s00440-023-01227-3","DOIUrl":"https://doi.org/10.1007/s00440-023-01227-3","url":null,"abstract":"Abstract The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the Lévy shifts of the sample paths. We answer to the affirmative Tsirelson’s question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener–Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134960043","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}