Probability Measure Research Articles

Depths in random recursive metric spaces

Abstract As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

Spectral structure of Moran Sierpinski-type measure on R2

Let Mn=diag[3pn,3qn] with pn,qn⩾1 for all n⩾1 and let D={(0,0)t,(1,0)t,(0,1)t} . One can generate a Borel probability measure μ{Mn},D=δM1−1D∗δ(M2M1)−1D∗δ(M3M2M1)−1D∗⋯. Such measure μ{Mn},D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L2(μ{Mn},D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L2(μ{Mn},D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L2(μ{Mn},D) or not.

The Wasserstein Distance for Ricci Shrinkers

Abstract Let $(M^{n},g,f)$ be a Ricci shrinker such that $\text{Ric}_{f}=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi )^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $\gamma $ and the measure $\overline{\nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $\overline{\nu }$ and $\gamma $, and which also elucidates the rigidity implications resulting from this estimate.

Percolation transition for random forests in $d\geqslant 3$

AbstractThe arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta >0$ β > 0 per edge. It arises as the $q\to 0$ q → 0 limit of the $q$ q -state random cluster model with $p=\beta q$ p = β q . We prove that in dimensions $d\geqslant 3$ d ⩾ 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ d = 2 where no percolation transition occurs.The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\mathbb{H}^{0|2}$ H 0 | 2 . This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the $\mathbb{H}^{0|2}$ H 0 | 2 model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

Trend to Equilibrium for Run and Tumble Equations with Non-uniform Tumbling Kernels

We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.

Finitely additive behavioral strategies: when do they induce an unambiguous expected payoff?

Finitely additive behavioral strategies: when do they induce an unambiguous expected payoff?

Spectral MV-algebras and equispectrality

AbstractIn this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both bipartite MV-algebras and spectral MV-algebras can be finitely axiomatized in first order logic. We also prove that there is only, up to isomorphism, a set of MV-algebras with given prime spectrum. A further part of the paper is devoted to some relations between bipartite MV-algebras and their states. Recall that a state on an MV-algebra is a generalization of a probability measure on a Boolean algebra. Particular states are the states with Bayes’ property. We show that an MV-algebra admits a state with the Bayes’ property if and only if it is bipartite.

Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we consider integral equations featuring kernels which may be expressed as scalar multiples of conservative (i.e. Markov) kernels and we adopt a variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean–Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

AbstractLet be a random matrix distributed according to uniform probability measure on the finite general linear group . We show that equidistributes on as as long as and that this range is sharp. We also show that nontrivial linear combinations of equidistribute as long as and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for , where depends on , due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of , we end up showing that certain explicit character sums modulo exhibit cancellation when averaged over monic polynomials of degree in as long as . This goes far beyond the classical range due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

P-adic Brownian motion is a scaling limit

A p-adic Brownian motion is a continuous time stochastic process in a p-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1,∞) , and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Magnetization switching driven by spin current in a T-type ferromagnetic trilayer

The T-type CoFeB/spacer/CoFeB structure is a promising candidate for the development of perpendicular spin–orbit torque (SOT) magnetic random-access memory and other SOT devices. It consists of an in-plane magnetized layer, a perpendicularly magnetized layer, and a non-magnetic metal spacer that induces interlayer exchange coupling. By engineering the W spacer, this system achieves field-free SOT switching with a nearly 100% switching ratio. Furthermore, it realizes a high exchange coupling field of 255 Oe using a relatively thinner spacer thickness, enhancing the reliability and energy efficiency of SOT switching. Measurement of current switching probability suggests that this perpendicular magnetic anisotropy system may enable the implementation of probability-adjustable true random number generators in future applications. The T-type structures with strong interlayer coupling exhibit great potential for spintronic device applications.

Spectrality of homogeneous Moran measures on the plane

LetM=ρ1−1a0ρ2−1 be a real upper triangular expanding matrix and D=00,d10,d2d3 be a three-element real digit set with d1d3≠0 , and let {nk}k=1∞ be a sequence of positive integers with upper bound. The infinite convolutions μM,D,{nk}=δM−n1D∗δM−(n1+n2)D∗⋯∗δM−(n1+⋯+nk)D∗⋯ converges weakly to a Borel probability measure (homogeneous Moran measure). In this paper, we study the existence of exponential orthonormal basis for L2(μM,D,{nk}). A necessary and sufficient condition for μM,D,{nk} to be a spectral measure is established.

On a novel gradient flow structure for the aggregation equation

The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous inelastic Boltzmann equation, a formal Taylor expansion reveals a link between this equation and the aggregation equation with an appropriately chosen interaction potential. Inspired by this formal link and the fact that the associated aggregation equation also dissipates the kinetic energy, we present a novel way of interpreting the aggregation equation as a gradient flow, in the sense of curves of maximal slope, of the kinetic energy, rather than the usual interaction energy, with respect to an appropriately constructed transportation metric on the space of probability measures.

Density estimation via measure transport: Outlook for applications in the biological sciences

AbstractOne among several advantages of measure transport methods is that they allow or a unified framework for processing and analysis of data distributed according to a wide class of probability measures. Within this context, we present results from computational studies aimed at assessing the potential of measure transport techniques, specifically, the use of triangular transport maps, as part of a workflow intended to support research in the biological sciences. Scenarios characterized by the availability of limited amount of sample data, which are common in domains such as radiation biology, are of particular interest. We find that when estimating a distribution density function given limited amount of sample data, adaptive transport maps are advantageous. In particular, statistics gathered from computing series of adaptive transport maps, trained on a series of randomly chosen subsets of the set of available data samples, leads to uncovering information hidden in the data. As a result, in the radiation biology application considered here, this approach provides a tool for generating hypotheses about gene relationships and their dynamics under radiation exposure.

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

Imprecision in martingale- and test-theoretic prequential randomness

In a prequential approach to algorithmic randomness, probabilities for the next outcome can be forecast ‘on the fly’ without the need for fully specifying a probability measure on all possible sequences of outcomes, as is the case in the more standard approach. We take the first steps in allowing for probability intervals instead of precise probabilities on this prequential approach, based on ideas borrowed from our earlier imprecise-probabilistic, standard account of algorithmic randomness. We define what it means for an infinite sequence (I1,x1,I2,x2,…) of successive interval forecasts Ik and subsequent binary outcomes xk to be random, both in a martingale-theoretic and a test-theoretic sense. We prove that these two versions of prequential randomness coincide, we compare the resulting prequential randomness notions with the more standard ones, and we investigate where the prequential and standard randomness notions coincide.

Surface recombination in Pyrex in oxygen DC glow discharges: mesoscopic modelling and comparison with experiments

Surface recombination in an oxygen DC glow discharge in a Pyrex (borosilicate glass) tube is studied via mesoscopic modelling and comparison with measurements of recombination probability. A total of 106 experimental conditions are assessed, with discharge current varying between 10 and 40 mA, pressure values ranging between 0.75 and 10 Torr, and fixed outer wall temperatures ( Tw ) of −20, 5, 25 and 50∘C . The model includes O+O and O+O2 surface recombination reactions and a Tw dependent desorption frequency. The model is validated for all the 106 studied conditions and intends to have predictive capabilities. The analysis of the simulation results highlights that for Tw=−20∘C and Tw=5∘C the dominant recombination mechanisms involve physisorbed oxygen atoms ( OF ) in Langmuir–Hinshelwood (L-H) recombination OF+OF and in Eley–Rideal (E-R) recombination O2+OF , while for Tw=25∘C and Tw=50∘C processes involving chemisorbed oxygen atoms ( OS ) in E-R O+OS and L-H OF+OS also play a relevant role. A discussion is taken on the relevant recombination mechanisms and on ozone wall production, with relevance for higher pressure regimes.

Comparison of Dual Antiplatelet Therapies for Minor, Nondisabling, Acute Ischemic Stroke: A Bayesian Network Meta-Analysis.

Dual antiplatelet therapy (DAPT) appears to be an effective treatment option for minor (nondisabling) acute ischemic stroke. This conclusion is based on trials that include both transient ischemic attack (TIA) and minor stroke; however, these 2 conditions may differ. To compare DAPT regimens specifically for minor stroke. PubMed was searched for randomized clinical trials published up to November 4, 2023. Search terms strategy included TIA, transient ischemic attack, minor stroke, or moderate stroke, with the filter randomized controlled trial. Unpublished data on minor stroke were sourced from authors and/or institutions. Trials testing DAPT within the first 24 hours of a minor stroke (defined as a National Institutes of Health Stroke Scale score ≤5) were included by consensus. Of 1508 studies screened, 6 (0.3%) initially met inclusion criteria and were reviewed. The study was performed according to Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines by multiple observers. Bayesian fixed-effect network meta-analysis was conducted. Secondary analysis performed for high-risk TIA alone. Treatments were ranked using a probability measure called surface under the cumulative rank curve (SUCRA). The primary outcome was subsequent ischemic stroke at 90 days. Secondary outcomes included major hemorrhage, mortality, and hemorrhagic stroke. The number needed to treat (NNT) and number needed to harm (NNH) were obtained. Five trials were included that described 28 148 patients, of whom 22 203 (78.9%) had a minor stroke. Of these, 13 995 (63.0%) were in DAPT groups and 8208 (37.0%) in aspirin (acetylsalicylic acid) groups. Aspirin and ticagrelor had a 94% probability of being the superior treatment for minor stroke (SUCRA, 0.94) for the primary outcome. Both aspirin and ticagrelor (NNT, 40; 95% CI, 31-64) and aspirin and clopidogrel (NNT, 58; 95% CI, 39-136) were superior to aspirin alone in the prevention of recurrent ischemic stroke at 90 days. Both treatments had higher rates of major hemorrhage than aspirin alone (NNH for aspirin and ticagrelor, 284; 95% CI, 108-1715 vs NNH for aspirin and clopidogrel, 330; 95% CI, 118-3430), but neither had increased risk of hemorrhagic stroke or death. For high-risk TIA, ticagrelor and aspirin had a 60% probability (SUCRA, 0.60) and clopidogrel and aspirin had a 40% probability (SUCRA 0.40) of being a superior treatment; neither was optimum, but both were superior to aspirin alone for the primary outcome. These findings suggest that DAPT with aspirin and ticagrelor has higher probability of being the superior treatment among patients with minor stroke when presence of CYP2C19 loss-of-function alleles has not been excluded. For patients with TIA, the superiority of aspirin and ticagrelor vs aspirin and clopidogrel was not demonstrated.