Mostrando recursos 1 - 20 de 74

  1. Intertwinings of beta-Dyson Brownian motions of different dimensions

    Ramanan, Kavita; Shkolnikov, Mykhaylo
    We show that for all positive $\beta$ the semigroups of $\beta$-Dyson Brownian motions of different dimensions are intertwined. The proof relates $\beta$-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption $\beta\ge1$ in (Probab. Theory Related Fields 163 (2015) 413–463). The corresponding results for $\beta$-Dyson Ornstein–Uhlenbeck processes are also presented.

  2. Universality in a class of fragmentation-coalescence processes

    Kyprianou, A. E.; Pagett, S. W.; Rogers, T.
    We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit a universal cluster size distribution regardless of the details of the rates, following a power law with exponent $3/2$.

  3. From optimal stopping boundaries to Rost’s reversed barriers and the Skorokhod embedding

    De Angelis, Tiziano
    We provide a new probabilistic proof of the connection between Rost’s solution of the Skorokhod embedding problem and a suitable family of optimal stopping problems for Brownian motion, with finite time-horizon. In particular we use stochastic calculus to show that the time reversal of the optimal stopping sets for such problems forms the so-called Rost’s reversed barrier.

  4. Variational multiscale nonparametric regression: Smooth functions

    Grasmair, Markus; Li, Housen; Munk, Axel
    For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski–Dantzig (MIND) estimator. This can be viewed as a multiscale extension of the Dantzig selector (Ann. Statist. 35 (2009) 2313–2351) based on early ideas of Nemirovski (J. Comput. System Sci. 23 (1986) 1–11). MIND minimizes a homogeneous Sobolev norm under the constraint that the multiresolution norm of the residual is bounded by a universal threshold. The main contribution of this paper is the derivation of convergence rates of MIND with respect to $L^{q}$-loss, $1\le q\le\infty $, both almost...

  5. Height fluctuations of stationary TASEP on a ring in relaxation time scale

    Liu, Zhipeng
    We consider the totally asymmetric simple exclusion process on a ring with stationary initial conditions. The crossover between KPZ dynamics and equilibrium dynamics occurs when time is proportional to the $3/2$ power of the ring size. We obtain the limit of the height function along the direction of the characteristic line in this time scale. The two-point covariance function in this scale is also discussed.

  6. Stochastic orders and the frog model

    Johnson, Tobias; Junge, Matthew
    The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. ¶ This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include...

  7. $\operatorname{ASEP}(q,j)$ converges to the KPZ equation

    Corwin, Ivan; Shen, Hao; Tsai, Li-Cheng
    We show that a generalized Asymmetric Exclusion Process called $\operatorname{ASEP}(q,j)$ introduced in (Probab. Theory Related Fields 166 (2016) 887–933). converges to the Cole–Hopf solution to the KPZ equation under weak asymmetry scaling.

  8. Rigidity of 3-colorings of the discrete torus

    Feldheim, Ohad Noy; Peled, Ron
    We prove that a uniformly chosen proper $3$-coloring of the $d$-dimensional discrete torus has a very rigid structure when the dimension $d$ is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the $3$-state anti-ferromagnetic Potts model from statistical physics. ¶ Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain...

  9. Non-fixation for biased Activated Random Walks

    Rolla, L. T.; Tournier, L.
    We prove that the model of Activated Random Walks on $\mathbb{Z}^{d}$ with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to $1$. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.

  10. Eta-diagonal distributions and infinite divisibility for R-diagonals

    Bercovici, Hari; Nica, Alexandru; Noyes, Michael; Szpojankowski, Kamil
    The class of $R$-diagonal $*$-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an $\eta$-diagonal distribution that is the Boolean counterpart of an $R$-diagonal distribution. We establish a number of properties of $\eta$-diagonal distributions, then we examine the canonical bijection relating $\eta$-diagonal distributions to infinitely divisible $R$-diagonal ones. The overall result is a parametrization of an arbitrary...

  11. Distributional limits of positive, ergodic stationary processes and infinite ergodic transformations

    Aaronson, Jon; Weiss, Benjamin
    In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable – and $\alpha$-rationally ergodic – transformations are presented.

  12. Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$

    Ziesche, Sebastian
    We consider the Boolean model $Z$ on $\mathbb{R}^{d}$ with random compact grains of bounded diameter, i.e. $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ where $\{X_{1},X_{2},\dots\}$ is a Poisson point process of intensity $t$ and $(Z_{1},Z_{2},\dots)$ is an i.i.d. sequence of compact grains (not necessarily balls) with diameters a.s. bounded by some constant. We will show that exponential decay holds in the sub-critical regime, that means the volume and radius of the cluster of the typical grain in $Z$ have an exponential tail. To achieve this we adapt the arguments of (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015)...

  13. Multidimensional two-component Gaussian mixtures detection

    Laurent, Béatrice; Marteau, Clément; Maugis-Rabusseau, Cathy
    Let $(X_{1},\ldots,X_{n})$ be a $d$-dimensional i.i.d. sample from a distribution with density $f$. The problem of detection of a two-component mixture is considered. Our aim is to decide whether $f$ is the density of a standard Gaussian random $d$-vector ($f=\phi_{d}$) against $f$ is a two-component mixture: $f=(1-\varepsilon)\phi_{d}+\varepsilon\phi_{d}(\cdot -\mu)$ where $(\varepsilon,\mu)$ are unknown parameters. Optimal separation conditions on $\varepsilon$, $\mu$, $n$ and the dimension $d$ are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two alternative subsets are considered.

  14. Degrees of freedom for piecewise Lipschitz estimators

    Mikkelsen, Frederik Riis; Hansen, Niels Richard
    A representation of the degrees of freedom akin to Stein’s lemma is given for a class of estimators of a mean value parameter in $\mathbb{R}^{n}$. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate of the...

  15. Monotonicity and condensation in homogeneous stochastic particle systems

    Rafferty, Thomas; Chleboun, Paul; Grosskinsky, Stefan
    We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On fixed finite...

  16. Stochastic Ising model with flipping sets of spins and fast decreasing temperature

    Cerqueti, Roy; De Santis, Emilio
    This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in $\mathbb{R}^{d}$. The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support $\{-1,+1\}$. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the...

  17. Stochastic integral equations for Walsh semimartingales

    Ichiba, Tomoyuki; Karatzas, Ioannis; Prokaj, Vilmos; Yan, Minghan
    We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison–Shepp-type equations and a change-of-variable formula in the spirit of Freidlin–Sheu for these so-called “Walsh semimartingales”. We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.

  18. Davie’s type uniqueness for a class of SDEs with jumps

    Priola, Enrico
    A result of A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) states that a multidimensional stochastic equation $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, driven by a Wiener process $W=(W_{t})$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Wiener path. We consider a similar problem when $W$ is replaced by a Lévy process $L=(L_{t})$ and $b$ is $\beta$-Hölder continuous in the space variable, $\beta\in(0,1)$. We assume that $L_{1}$ has a finite moment of order $\theta$, for some ${\theta}>0$. Using a new càdlàg regularity result for strong solutions, we prove that strong...

  19. KPZ and Airy limits of Hall–Littlewood random plane partitions

    Dimitrov, Evgeni
    In this paper we consider a probability distribution $\mathbb{P}^{q,t}_{\mathrm{HL}}$ on plane partitions, which arises as a one-parameter generalization of the standard $q^{\mathrm{volume}}$ measure. This generalization is closely related to the classical multivariate Hall–Littlewood polynomials, and it was first introduced by Vuletić in (Trans. Am. Math. Soc. 361 (2009) 2789–2804). ¶ We prove that as the plane partitions become large ($q$ goes to $1$, while the Hall–Littlewood parameter $t$ is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy–Widom distribution....

  20. Perturbation by non-local operators

    Chen, Zhen-Qing; Wang, Jie-Ming
    Suppose that $d\ge1$ and $0<\beta<\alpha<2$. We establish the existence and uniqueness of the fundamental solution $q^{b}(t,x,y)$ to a class of (typically non-symmetric) non-local operators $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$, where ¶ \[\mathcal{S}^{b}f(x):=\mathcal{A}(d,-\beta)\int_{\mathbb{R}^{d}}(f(x+z)-f(x)-\nabla f(x)\cdot z\mathbb{1}_{\{|z|\leq1\}})\frac{b(x,z)}{|z|^{d+\beta}}\,dz\] and $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $b(x,z)=b(x,-z)$ for $x,z\in\mathbb{R}^{d}$. Here $\mathcal{A}(d,-\beta)$ is a normalizing constant so that $\mathcal{S}^{b}=\Delta^{\beta/2}$ when $b(x,z)\equiv1$. We show that if $b(x,z)\geq-\frac{\mathcal{A}(d,-\alpha)}{\mathcal{A}(d,-\beta)}|z|^{\beta-\alpha}$, then $q^{b}(t,x,y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^{b}$, which has strong Feller property. The Feller process $X^{b}$ is the unique solution to the martingale problem of $(\mathcal{L}^{b},\mathcal{S}(\mathbb{R}^{d}))$, where $\mathcal{S}(\mathbb{R}^{d})$ denotes the space of...

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