Probability Theory and Related Fields最新文献

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.

We prove the Perelman type W-entropy formula for the geodesic flow on the (L^2)-Wasserstein space over a complete Riemannian manifold equipped with Otto’s infinite dimensional Riemannian metric. To better understand the similarity between the W-entropy formula for the geodesic flow on the Wasserstein space and the W-entropy formula for the heat flow of the Witten Laplacian on the underlying manifold, we introduce the Langevin deformation of flows on the Wasserstein space over a Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over a Riemannian manifold, and can be regarded as the potential flow of the compressible Euler equation with damping on a Riemannian manifold. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and a compact Riemannian manifold, and prove the convergence of the Langevin deformation for (crightarrow 0) and (crightarrow infty ) respectively. Moreover, we prove the W-entropy-information formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. The rigidity theorems are proved for the W-entropy for the geodesic flow and the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the CD(0, m)-condition. Our results are new even in the case of Euclidean spaces and complete Riemannian manifolds with non-negative Ricci curvature.

We consider the problem of finding the initial vertex (Adam) in a Barabási–Albert tree process ( (mathcal {T}(n): n ge 1)) at large times. More precisely, given ( varepsilon >0), one wants to output a subset ( mathcal {P}_{ varepsilon }(n)) of vertices of ( mathcal {T}(n)) so that the initial vertex belongs to ( mathcal {P}_ varepsilon (n)) with probability at least (1- varepsilon ) when n is large. It has been shown by Bubeck, Devroye and Lugosi, refined later by Banerjee and Huang, that one needs to output at least ( varepsilon ^{-1 + o(1)}) and at most (varepsilon ^{-2 + o(1)}) vertices to succeed. We prove that the exponent in the lower bound is sharp and the key idea is that Adam is either a “large degree" vertex or is a neighbor of a “large degree" vertex (Eve).

We consider the standard first passage percolation model on ({mathbb {Z}}^ d) with a distribution G taking two values (0<a<b). We study the maximal flow through the cylinder ([0,n]^ {d-1}times [0,hn]) between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in (O(frac{n^{d-1}}{log n})), for (hge h_0) (for a large enough constant (h_0=h_0(a,b))). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder ([0,n]^ {d-1}times [0,hn]) is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant (hge h_0) (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini–Kalai–Schramm (Ann Probab 31:1970–1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970–1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.

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 in (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):550–574, 2007) and the one for Brownian motion ((H = 1/2)) (Fawcett 2003), to the general case (H > 1/4), thereby resolving an established open problem. Other processes studied include continuous and centred Gaussian semimartingales.

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 (nu ^{m}) is quantitatively well-approximated by (big (text {Id},exp (nabla h^{n})big )_#mu ^n) where (h^{n}) solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge–Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the (alpha )-mixing coefficient holds and for a class of discrete-time sub-geometrically ergodic Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure.

Abstract

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in Komorowski et al. (Ann Prob 48:2290–2322, 2020), where the case of a non-degenerate probability of killing has been considered.

Associated to two given sequences of eigenvalues (lambda _1 ge cdots ge lambda _n) and (mu _1 ge cdots ge mu _n) is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as (n rightarrow infty ). Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process.