Probability Theory and Related Fields最新文献

We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the N-spin system to that of suitably conditioned subsystems.

In this paper, we study the online nearest neighbor random tree in dimension (din {mathbb {N}}) (called d-NN tree for short) defined as follows. We fix the torus ({mathbb {T}}^d_n) of dimension d and area n and equip it with the metric inherited from the Euclidean metric in ({mathbb {R}}^d). Then, embed consecutively n vertices in ({mathbb {T}}^d_n) uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph (G_n). We show multiple results concerning the degree sequence of (G_n). First, we prove that typically the number of vertices of degree at least (kin {mathbb {N}}) in the d-NN tree decreases exponentially with k and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of (G_n) is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in (G_n) is ((1+o(1))log n) and the diameter of ({mathbb {T}}^d_n) is ((2e+o(1))log n), independently of the dimension. Finally, we define a natural infinite analog (G_{infty }) of (G_n) and show that it corresponds to the local limit of the sequence of finite graphs ((G_n)_{n ge 1}). Furthermore, we prove almost surely that (G_{infty }) is locally finite, that the simple random walk on (G_{infty }) is recurrent, and that (G_{infty }) is connected.

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if (textsf{L}_N) is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then (tfrac{pi }{log N} ,textsf{L}_N) converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if (textsf{L}_N^{(1,2)}=sum _{n=1}^N mathbb {1}_{{S_n^{(1)}= S_n^{(2)}}}), then (tfrac{pi }{log N}, textsf{L}^{(1,2)}_N) converges in distribution to an exponential random variable of parameter one. We prove that for every (h geqslant 3), the family ( big { frac{pi }{log N} ,textsf{L}_N^{(i,j)} big }_{1leqslant i<jleqslant h}), of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in ({mathbb {R}}^n) with a quadratic, conservative nonlinearity B(x, x) and a linear damping term—Ax which is degenerate in the sense that (textrm{ker} A ne emptyset ). We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if A is full rank, but otherwise, energy could potentially accumulate in (textrm{ker} A) and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of (textrm{ker} A) and many examples where such estimates can be made.

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math 147(1-2):71–88, 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 336:145, 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino’s result (Probab Theory Relat Fields 156:739–793, 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form.

Abstract

We consider the following stochastic heat equation $$begin{aligned} partial _t u(t,x) = tfrac{1}{2} partial ^2_x u(t,x) + b(u(t,x)) + sigma (u(t,x)) {dot{W}}(t,x), end{aligned}$$ defined for ((t,x)in (0,infty )times {mathbb {R}}) , where ({dot{W}}) denotes space-time white noise. The function (sigma ) is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition $$begin{aligned} int _1^infty frac{textrm{d}y}{b(y)}<infty end{aligned}$$ implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that (textrm{P}{ u(t,x)=infty quad hbox { for all } t>0 hbox { and } xin {mathbb {R}}}=1.) The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022).

We investigate the quadratic Schrödinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schrödinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schrödinger bridges satisfy a logarithmic Sobolev inequality on the product space. Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton–Jacobi–Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate (mu ) between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order (n^2/mu ). In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov’s large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko–Cantelli theorems, and estimates of weighted Poincaré–Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincaré–Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Braunstein et al. in Int J Bifurc Chaos 17(07):2215–2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006; Wu et al. in Phys Rev Lett 96(14):148702, 2006) is as follows: for a large class of random graph models with degree exponent (tau in (3,4)), distances in the minimal spanning tree (MST) on the giant component in the supercritical regime scale like (n^{(tau -3)/(tau -1)}). The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent (tau in (3, 4)) that is closely related to Aldous’s multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by (n^{-(tau -3)/(tau -1)}), converges in distribution with respect to the Gromov–Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals ((tau -1)/(tau -3)). The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.