Probability Theory and Related Fields最新文献

{"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":"8 1","pages":""},"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}

{"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":"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}

{"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":"<p>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 <i>typical</i> square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random <span>(ntimes n)</span> standard Gaussian coefficient matrix <i>A</i>, the <i>growth factor</i> of the Gaussian elimination with partial pivoting is at most polynomially large in <i>n</i> with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve <span>(Ax = b)</span> to <i>m</i> bits of accuracy using GEPP is <span>(m+O(log n))</span>, which improves an earlier estimate <span>(m+O(log ^2 n))</span> 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</p>","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}

{"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":"<p>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 <i>private measure</i> 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 <span>(varepsilon )</span> bounded away from zero. A key ingredient in our construction is a new <i>superregular random walk</i>, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly.\u0000</p>","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}

{"title":"Dirichlet forms on unconstrained Sierpinski carpets","authors":"Shiping Cao, Hua Qiu","doi":"10.1007/s00440-024-01280-6","DOIUrl":"https://doi.org/10.1007/s00440-024-01280-6","url":null,"abstract":"<p>We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the 1/<i>k</i> grids. The intersection of two cells can be a line segment of irrational length, and the non-diagonal assumption is dropped in this recurrent setting.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"41 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591636","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":"Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise","authors":"Viorel Barbu, Michael Röckner","doi":"10.1007/s00440-024-01277-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01277-1","url":null,"abstract":"<p>This work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator <span>((- Delta )^s)</span> for <span>(sin left( frac{1}{2},1right) )</span>. The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean–Vlasov equations with Lévy noise, as well as the Markov property for their laws are proved.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"12 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591630","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 structure of the local time of Markov processes indexed by Lévy trees","authors":"Armand Riera, Alejandro Rosales-Ortiz","doi":"10.1007/s00440-023-01258-w","DOIUrl":"https://doi.org/10.1007/s00440-023-01258-w","url":null,"abstract":"<p>We construct an additive functional playing the role of the local time—at a fixed point <i>x</i>—for Markov processes indexed by Lévy trees. We start by proving that Markov processes indexed by Lévy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct our additive functional by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Markov process takes the value <i>x</i> encode a new Lévy tree and we construct explicitly its height process. In particular, we recover a recent result of Le Gall concerning the subordinate tree of the Brownian tree where the subordination function is given by the past maximum process of Brownian motion indexed by the Brownian tree.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"45 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592141","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":"A Ray–Knight theorem for $$nabla phi $$ interface models and scaling limits","authors":"Jean-Dominique Deuschel, Pierre-François Rodriguez","doi":"10.1007/s00440-024-01275-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01275-3","url":null,"abstract":"<p>We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is <i>not</i> Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on <span>(mathbb R^3)</span> with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"51 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591548","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":"Diophantine conditions in the law of the iterated logarithm for lacunary systems","authors":"Christoph Aistleitner, Lorenz Frühwirth, Joscha Prochno","doi":"10.1007/s00440-024-01272-6","DOIUrl":"https://doi.org/10.1007/s00440-024-01272-6","url":null,"abstract":"<p>It is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erdős and Fortet in the 1950s that probability theory’s limit theorems may fail for lacunary sums <span>(sum f(n_k x))</span> if the sequence <span>((n_k)_{k ge 1})</span> has a strong arithmetic “structure”. The presence of such structure can be assessed in terms of the number of solutions <span>(k,ell )</span> of two-term linear Diophantine equations <span>(a n_k - b n_ell = c)</span>. As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the Erdős–Fortet example, and guarantees that <span>(sum f(n_k x))</span> satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure “truly independent” behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of <span>((n_k)_{k ge 1})</span> to ensure that <span>(sum f(n_k x))</span> shows “truly random” behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"52 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591556","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":"Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay","authors":"Rodrigo Bazaes, Isabel Lammers, Chiranjib Mukherjee","doi":"10.1007/s00440-024-01271-7","DOIUrl":"https://doi.org/10.1007/s00440-024-01271-7","url":null,"abstract":"<p>We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if <span>(H_T(omega ))</span> is a random field defined w.r.t. space-time white noise <span>(dot{B})</span> and integrated w.r.t. Brownian paths in <span>(dge 3)</span>, we consider the renormalized exponential <span>(mu _{gamma ,T})</span>, weighted w.r.t. the Wiener measure <span>(mathbb {P}_0(textrm{d}omega ))</span>. We construct the almost sure limit <span>(mu _gamma = lim _{Trightarrow infty } mu _{gamma ,T})</span> in the <i>entire weak disorder (subcritical)</i> regime and call it <i>subcritical GMC on the Wiener space</i>. We show that </p><span>$$begin{aligned} mu _gamma Big {omega : lim _{Trightarrow infty } frac{H_T(omega )}{T(phi star phi )(0)} ne gamma Big }=0 qquad text{ almost } text{ surely, } end{aligned}$$</span><p>meaning that <span>(mu _gamma )</span> is supported almost surely only on <span>(gamma )</span>-<i>thick paths</i>, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit <span>(mu _gamma )</span> w.r.t. the mollification scheme <span>(phi )</span> in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of <span>(dot{B})</span> under the random <i>rooted</i> measure <span>(mathbb Q_{mu _gamma }(textrm{d}dot{B}textrm{d}omega )= mu _gamma (textrm{d}omega ,dot{B})P(textrm{d}dot{B}))</span> is the same as the law of the distribution <span>(fmapsto dot{B}(f)+ gamma int _0^infty int _{mathbb {R}^d} f(s,y) phi (omega _s-y) textrm{d}s textrm{d}y)</span> under <span>(P otimes mathbb {P}_0)</span>. We then determine the fractal properties of the measure around <span>(gamma )</span>-thick paths: <span>(-C_2 le liminf _{varepsilon downarrow 0} varepsilon ^2 log {widehat{mu }}_gamma (Vert omega Vert&lt; varepsilon ) le limsup _{varepsilon downarrow 0}sup _eta varepsilon ^2 log {widehat{mu }}_gamma (Vert omega -eta Vert &lt; varepsilon ) le -C_1)</span> w.r.t a weighted norm <span>(Vert cdot Vert )</span>. Here <span>(C_1&gt;0)</span> and <span>(C_2&lt;infty )</span> are the uniform upper (resp. pointwise lower) Hölder exponents which are <i>explicit</i> in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and <span>(L^p)</span> (<span>(p&gt;1)</span>) moments for the total mass of <span>(mu _gamma )</span> in the weak disorder regime.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"12 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140591528","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}