Probability Theory and Related Fields最新文献

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.

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

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 _textsf {RW}) of the lazy random walk on G is characterised by the edge conductance (Phi ) of G via Cheeger’s inequality: (Phi ^{-1} lesssim tau _textsf {RW} lesssim Phi ^{-2} log |V|). Analogously, we characterise the fastest mixing time (tau ^star ) via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance (Psi ) of G: (Psi ^{-1} lesssim tau ^star lesssim Psi ^{-2} (log |V|)^2). This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only (varepsilon )-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time (tau lesssim varepsilon ^{-1} ({text {diam}} G)^2 log |V|). Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.

In the negative perceptron problem we are given n data points ((varvec{x}_i,y_i)), where (varvec{x}_i) is a d-dimensional vector and (y_iin {+1,-1}) is a binary label. The data are not linearly separable and hence we content ourselves to find a linear classifier with the largest possible negative margin. In other words, we want to find a unit norm vector (varvec{theta }) that maximizes (min _{ile n}y_ilangle varvec{theta },varvec{x}_irangle ). This is a non-convex optimization problem (it is equivalent to finding a maximum norm vector in a polytope), and we study its typical properties under two random models for the data. We consider the proportional asymptotics in which (n,drightarrow infty ) with (n/drightarrow delta ), and prove upper and lower bounds on the maximum margin (kappa _{{textrm{s}}}(delta )) or—equivalently—on its inverse function (delta _{{textrm{s}}}(kappa )). In other words, (delta _{{textrm{s}}}(kappa )) is the overparametrization threshold: for (n/dle delta _{{textrm{s}}}(kappa )-{varepsilon }) a classifier achieving vanishing training error exists with high probability, while for (n/dge delta _{{textrm{s}}}(kappa )+{varepsilon }) it does not. Our bounds on (delta _{{textrm{s}}}(kappa )) match to the leading order as (kappa rightarrow -infty ). We then analyze a linear programming algorithm to find a solution, and characterize the corresponding threshold (delta _{textrm{lin}}(kappa )). We observe a gap between the interpolation threshold (delta _{{textrm{s}}}(kappa )) and the linear programming threshold (delta _{textrm{lin}}(kappa )), raising the question of the behavior of other algorithms.