Theory Probab Research Articles

Mosco Convergence of Gradient Forms with Non-Convex Interaction Potential

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, EN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {E}}^N$$\\end{document} on L2(E,μN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(E,\\mu _N)$$\\end{document} for N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in {\\mathbb {N}}$$\\end{document}, in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} on the state space E—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} try to impose as little restrictions as possible. The problem has fully been solved if the family (μN)N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${(\\mu _N)}_{N}$$\\end{document} contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).

Refined Universality for Critical KCM: Upper Bounds

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions. They are tightly linked to the monotone cellular automata called bootstrap percolation. Among the three classes of such models (Bollobás et al. in Combin Probab Comput 24(4):687–722, 2015), the critical ones are the most studied. Together with the companion paper by Marêché and the author (Hartarsky and Marêché in Combin Probab Comput 31(5):879–906, 2022), our work determines the logarithm of the infection time up to a constant factor for all critical KCM. This was previously known only up to logarithmic corrections (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). We establish that on this level of precision critical KCM have to be classified into seven categories. This refines the two classes present in bootstrap percolation (Bollobás et al. in Proc Lond Math Soc (3) 126(2):620–703, 2023) and the two in previous rougher results (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). In the present work we establish the upper bounds for the novel five categories and thus complete the universality program for equilibrium critical KCM. Our main innovations are the identification of the dominant relaxation mechanisms and a more sophisticated and robust version of techniques recently developed for the study of the Fredrickson-Andersen 2-spin facilitated model (Hartarsky et al. in Probab Theory Relat Fields 185(3):993–1037, 2023).

On the Spectral Form Factor for Random Matrices

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Stable Random Variables with Complex Stability Index, II

This paper, which is a continuation of [I. A. Alexeev, Theory Probab. Appl., 67 (2022), pp. 335--351], is concerned with $\alpha$-stable distributions with complex stability index $\alpha$. Sufficient conditions for membership in the domain of attraction of $\alpha$-stable random variables (r.v.'s) are given, and $\alpha$-stable Lévy processes and the corresponding semigroups of operators are constructed. Necessary and sufficient conditions are given for membership in the class of limit laws for sums of independent and identically distributed (i.i.d.) complex r.v.'s with complex normalization and centering.

Distributed-order space-time fractional diffusions in bounded domains

We consider the distributed-order space-time fractional heat type equation \begin{document}$ \mathbb{D}^{(\nu)}u(t, x) = -(-\Delta)^{\alpha/2}u(t, x), \ t>0, \ x\in D, $\end{document} where$ \mathbb{D}^{(\nu)} $ is the distributed-order derivative based on the Caputo time-fractional derivative, $ -(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process for $ \alpha\in(0, 2] $, $ D $ is a bounded domain in $ \mathbb{R}^d $ and $ \nu $ is a finite Borel measure, known as mixing measure. An important application of distributed-order diffusions is to model ultraslow diffusions where a plume of particles spreads at a logarithmic rate as described in Sinai (Theory Probab. Appl., 27 (1982) 256–268). Using analytical tools, we provide an explicit classical solution and a stochastic analogue for this equation in bounded domains with zero exterior boundary conditions. We also show that our results still hold when the mixing measure in the distributed-order time-derivative is singular. Our results extend the results in the case of the time-fractional distributed-order diffusions obtained in Naber (Fractals 12(2004) 23–32) and Meerschaert et al. (J. Math. Anal. Appl., 379(2011) 216–228).

Letter to the Editors

This letter presents a correction to the paper [S. V. Nagaev, An alternative method of the proof of the ergodic theorem for general Markov chains, Theory Probab. Appl., 66 (2021), pp. 364--375].

Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper (Lee and Seo in Probab Theory Relat Fields 182:849–903, 2022), we proved the Eyring–Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of (Rezakhanlou and Seo in https://arxiv.org/abs/1812.02069, 2018), which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.

On a new concept of stochastic domination and the laws of large numbers

Consider a sequence of positive integers \(\{k_n,n\ge 1\}\), and an array of nonnegative real numbers \(\{a_{n,i},1\le i\le k_n,n\ge 1\}\) satisfying \(\sup _{n\ge 1}\sum _{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty ).\) This paper introduces the concept of \(\{a_{n,i}\}\)-stochastic domination. We develop some techniques concerning this concept and apply them to remove an assumption in a strong law of large numbers of Chandra and Ghosal (Acta Math Hung 71(4):327–336, 1996). As a by-product, a considerable extension of a recent result of Boukhari (J Theor Probab, 2021. https://doi.org/10.1007/s10959-021-01120-6) is established and proved by a different method. The results on laws of large numbers are new even when the summands are independent. Relationships between the concept of \(\{a_{n,i}\}\)-stochastic domination and the concept of \(\{a_{n,i}\}\)-uniform integrability are presented. Two open problems are also discussed.

Optimal robust mean and location estimation via convex programs with respect to any pseudo-norms

We consider the problem of robust mean and location estimation with respect to any pseudo-norm of the form\(x\in {{\mathbb {R}}}^d\mapsto \left\| x\right\| _S = \sup _{v\in S}\bigl < v,x \bigr>\) where S is any symmetric subset of \({{\mathbb {R}}}^d\). We show that the deviation-optimal minimax sub-Gaussian rate for confidence \(1-\delta \) is $$\begin{aligned} \max \left( \frac{\ell ^*(\Sigma ^{1/2}S)}{\sqrt{N}}, \sup _{v\in S}\left\| \Sigma ^{1/2}v\right\| _2\sqrt{\frac{\log (1/\delta )}{N}}\right) \end{aligned}$$where \(\ell ^*(\Sigma ^{1/2}S)\) is the Gaussian mean width of \(\Sigma ^{1/2}S\) and \(\Sigma \) the covariance of the data. This improves the entropic minimax lower bound from Lugosi and Mendelson (Probab Theory Relat Fields 175(3–4):957–973, 2019) and closes the gap characterized by Sudakov’s inequality between the entropy and the Gaussian mean width for this problem. This shows that the right statistical complexity measure for the mean estimation problem is the Gaussian mean width. We also show that this rate can be achieved by a solution to a convex optimization problem in the adversarial and \(L_2\) heavy-tailed setup by considering minimum of some Fenchel–Legendre transforms constructed using the median-of-means principle. We finally show that this rate may also be achieved in situations where there is not even a first moment but a location parameter exists.

Compactness and fractal dimensions of inhomogeneous continuum random trees

We introduce a new stick-breaking construction for inhomogeneous continuum random trees. This new construction allows us to prove the necessary and sufficient condition for compactness conjectured by Aldous et al. (Probab Theory Relat Fields 129(2):182–218, 2004) by comparison with Lévy trees. We also compute the fractal dimensions (Minkowski, Packing, Hausdorff).

Speed of Excited Random Walks with Long Backward Steps

We study a model of multi-excited random walk with non-nearest neighbour steps on mathbb {Z}, in which the walk can jump from a vertex x to either x+1 or x-i with iin {1,2,dots ,L}, Lge 1. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton–Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh (Probab Theory Relat Fields 141:3–4, 2008), we extend their result (w.r.t. the case L=1) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift delta >2. This confirms a special case of a conjecture proposed by Davis and Peterson.

Deviation results for sparse tables in hashing with linear probing

We consider the model of hashing with linear probing and we establish the moderate and large deviations for the total displacement in sparse tables. In this context, Weibull-like-tailed random variables appear. Deviations for sums of such heavy-tailed random variables are studied in Nagaev (Theory Probab Appl 14(1):51–64, 1969; Theory Probab Appl 14(2):193–208, 1969). Here we adapt the proofs therein to deal with conditioned sums of such variables and solve the open question in Gamboa et al. (Bernoulli 18(4):1341–1360, 2012). By the way, we establish the deviations of the total displacement in full tables, which can be derived from the deviations of empirical processes of i.i.d. random variables established in Wu (Ann Probab 22(1):17–27, 1994).

Reflecting Lévy Processes and Associated Families of Linear Operators. II

We consider special one-dimensional Markov processes, namely, asymmetric jump Lévy processes, which have values in a given interval and reflect from the boundary points. We show that in this case, in addition to the standard semigroup of operators generated by the Markov process, there also appears the family of “boundary” random operators that send functions defined on the boundary of the interval to elements of the space $L_2$ on the entire interval. This study is a continuation of our paper [Theory Probab. Appl., 64 (2019), 335--354], where a similar problem was solved for symmetric reflecting Lévy processes.

Growth-fragmentation process embedded in a planar Brownian excursion

The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane $${\mathbb {H}}$$ , obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in (Bertoin et al. Probab Theory Relat Field 172:663–724, 2018).

Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for q-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpreted as a q-discretization of the Beta polymer model from (Probab Theory Relat Fields 167(3):1057–1116 (2017). arXiv:1503.04117) with a new family of parameters, also attached to diagonals. To demonstrate how integrability with respect to the new diagonal parameters works, we extend the known results about Tracy–Widom large-scale fluctuations of the Beta polymer model.

Spin q-Whittaker Polynomials and Deformed Quantum Toda

Spin q-Whittaker symmetric polynomials labeled by partitions \(\lambda \) were recently introduced by Borodin and Wheeler (Spin q-Whittaker Polynomials, 2017. arXiv preprint arXiv:1701.06292 [math.CO]) in the context of integrable \(\mathfrak {sl}_2\) vertex models. They are a one-parameter deformation of the \(t=0\) Macdonald polynomials. We present a new more convenient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, \(q^{-\lambda _1}\) and \(q^{\lambda _N}\) (where \(\lambda \) is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP. In a scaling limit as \(q\nearrow 1\), spin q-Whittaker polynomials turn into a new one-parameter deformation of the \(\mathfrak {gl}_n\) Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as \(q\nearrow 1\) we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016. arXiv:1503.04117 [math.PR]), and relate it to spin Whittaker functions.

Chemical Subdiffusivity of Critical 2D Percolation

We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (Probab Theory Related Fields 73(3): 369–394 1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multiscale way of covering the graph so that “deep patches” have “thin backbones.” Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents \(\eta _2> \eta _1 > 0\): For d-dimensional models, the mean chemical displacement after T steps of random walk scales asymptotically slower than \(T^{1/\beta }\), whenever $$\begin{aligned} \beta < 2 + \frac{\eta _2-\eta _1}{d-\eta _1}\,. \end{aligned}$$Using the conjectured values of \(\eta _2 = \eta _1 + 1/4\) and \(\eta _1 = 5/48\) for 2D lattices, the latter quantity is \(2+12/91\).

Criticality and Covered Area Fraction in Confetti and Voronoi Percolation

Using the randomized algorithm method developed by Duminil-Copin et al. (Probab Theory Relat Fields 173(1–2):479–90, 2019), we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof, than that of Ahlberg et al. (Probab Theory Relat Fields 172(1–2):525–581, 2018), for the critical parameter for percolation in this model to be 1/2 when the radius of the underlying shapes for the distinct colours arise from the same distribution. In addition, we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain ‘transitivity condition’, this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.

Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula

In this article, we investigate the convergence rate of the discrete-time Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the \(L_{2}\)-convergence rate of the first-order error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”