Probability Theory and Related Fields

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• Private measures, random walks, and synthetic data AI翻译
  作者：Boedihardjo, March; Strohmer, Thomas; Vershynin, Roman
  作者单位：Michigan State University; University of California System; University of California Davis; University of California System; University of California Irvine
  摘要：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 clas...
• Geometric bounds on the fastest mixing Markov chain AI翻译
  作者：Olesker-Taylor, Sam; Zanetti, Luca
  作者单位：University of Warwick; University of Bath
  摘要：In the Fastest Mixing Markov Chain problem, we are given a graph G = (V, E) and desire the discrete-time Markov chain with smallest mixing time tau subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time tau(RW) of the lazy random walk on G is characterised by the edge conductance Phi of G via Cheeger's inequality: Phi(-1) less than or similar to tau(RW) less than or similar to Phi(-2) l...
• On the Wiener chaos expansion of the signature of a Gaussian process AI翻译
  作者：Cass, Thomas; Ferrucci, Emilio
  作者单位：Imperial College London; University of Oxford
  摘要：We compute the Wiener chaos decomposition of the signature for a class of Gaussian processes, which contains fractional Brownian motion (fBm) with Hurst parameter H is an element of (1/4,1). At level 0, our result yields an expression for the expected signature of such processes, which determines their law (Chevyrev and Lyons in Ann Probab 44(6):4049-4082, 2016). In particular, this formula simultaneously extends both the one for 1/2 < H-fBm (Baudoin and Coutin in Stochast Process Appl 117(5):...
• Stationary local random countable sets over the Wiener noise AI翻译
  作者：Vidmar, Matija; Warren, Jon
  作者单位：University of Ljubljana; University of Warwick
  摘要：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 Levy 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. ...
• CLT for NESS of a reaction-diffusion model AI翻译
  作者：Goncalves, P.; Jara, M.; Marinho, R.; Menezes, O.
  作者单位：Universidade de Lisboa; Universidade Federal de Santa Maria (UFSM); Universidade Federal da Bahia
  摘要：We study the scaling properties of the non-equilibrium stationary states (NESS) of a reaction-diffusion model. Under a suitable smallness condition, we show that the density of particles satisfies a law of large numbers with respect to the NESS, with an explicit rate of convergence, and we also show that at mesoscopic scales the NESS is well approximated by a local equilibrium (product) measure, in the total variation distance. In addition, in dimensions d <= 3\documentclass[12pt]{minimal} \us...
• The structure of the local time of Markov processes indexed by Lévy trees AI翻译
  作者：Riera, Armand; Rosales-Ortiz, Alejandro
  作者单位：Universite Paris Cite; Sorbonne Universite; University of Zurich
  摘要：We construct an additive functional playing the role of the local time-at a fixed point x-for Markov processes indexed by Levy trees. We start by proving that Markov processes indexed by Levy 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 o...
• Annealed quantitative estimates for the quadratic 2D-discrete random matching problem AI翻译
  作者：Clozeau, Nicolas; Mattesini, Francesco
  作者单位：Institute of Science & Technology - Austria; University of Munster; Max Planck Society
  摘要：We study a random matching problem on closed compact 2-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m = m(n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures mu(n) and v(m) is quantitat...
• Harnack inequality and one-endedness of UST on reversible random graphs AI翻译
  作者：Berestycki, Nathanael; van Engelenburg, Diederik
  作者单位：University of Vienna
  摘要：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...
• Fast-oscillating random perturbations of Hamiltonian systems AI翻译
  作者：Yan, Shuo
  作者单位：University System of Maryland; University of Maryland College Park
  摘要：We consider coupled slow-fast stochastic processes, where the averaged slow motion is given by a two-dimensional Hamiltonian system with multiple critical points. On a proper time scale, the evolution of the first integral converges to a diffusion process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices, as in the case of additive white noise perturbations of Hamiltonian systems considered by M. Freidlin and A. Wentzell. The current paper provi...
• Inhomogeneous long-range percolation in the weak decay regime AI翻译
  作者：Moench, Christian
  作者单位：Johannes Gutenberg University of Mainz
  摘要：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 weak decay regime, 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 tra...