Mostrando recursos 1 - 20 de 556

  1. Homogenization via sprinkling

    Benjamini, Itai; Tassion, Vincent
    We show that a superposition of an $\varepsilon$-Bernoulli bond percolation and any everywhere percolating subgraph of $\mathbb{Z}^{d}$, $d\ge2$, results in a connected subgraph, which after a renormalization dominates supercritical Bernoulli percolation. This result, which confirms a conjecture from (J. Math. Phys. 41 (2000) 1294–1297), is mainly motivated by obtaining finite volume characterizations of uniqueness for general percolation processes.

  2. Strong stationary times for one-dimensional diffusions

    Miclo, Laurent
    A necessary and sufficient condition is obtained for the existence of strong stationary times for ergodic one-dimensional diffusions, whatever the initial distribution. The strong stationary times are constructed through intertwinings with dual processes, in the Diaconis–Fill sense, taking values in the set of segments of the extended line $\mathbb{R}\sqcup\{-\infty,+\infty\}$. They can be seen as natural Doob transforms of the extensions to the diffusion framework of the evolving sets of Morris–Peres. Starting from a singleton set, the dual process begins by evolving into true segments in the same way a Bessel process of dimension 3 escapes from 0. The strong stationary...

  3. Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

    Zhu, Jiahui; Brzeźniak, Zdzisław; Hausenblas, Erika
    We consider a Banach space $(E,\|\cdot\|)$ such that, for some $q\geq2$, the function $x\mapsto\|x\|^{q}$ is of $C^{2}$ class and its $k$th, $k=1,2$, Fréchet derivatives are bounded by some constant multiples of the $(q-k)$th power of the norm. We also consider a $C_{0}$-semigroup $S$ of contraction type on $(E,\|\cdot\|)$. Finally we consider a compensated Poisson random measure $\tilde{N}$ on a measurable space $(Z,\mathcal{Z})$. ¶ We study the following stochastic convolution process ¶ \[u(t)=\int_{0}^{t}\!\int_{Z}S(t-s)\xi(s,z)\tilde{N}(\mathrm{d}s,\mathrm{d} z),\quad t\geq0,\] where $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes\mathcal{Z}$-predictable function. ¶ We prove that there exists a càdlàg modification $\tilde{u}$ of the process $u$ which satisfies the following maximal type inequality ¶ \[\mathbb{E}\sup_{0\leq s\leq t}\|\tilde{u}(s)\|^{q^{\prime}}\leq C\mathbb{E}(\int_{0}^{t}\!\int_{Z}\|\xi(s,z)\|^{p}N(\mathrm{d}s,\mathrm{d}z))^{\frac{q^{\prime}}{p}},\]...

  4. Unimodality for free Lévy processes

    Hasebe, Takahiro; Sakuma, Noriyoshi
    We will prove that: (1) A symmetric free Lévy process is unimodal if and only if its free Lévy measure is unimodal; (2) Every free Lévy process with boundedly supported Lévy measure is unimodal in sufficiently large time. (2) is completely different property from classical Lévy processes. On the other hand, we find a free Lévy process such that its marginal distribution is not unimodal for any time $s>0$ and its free Lévy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Lévy measure in (2) cannot be dropped. For the...

  5. Scaling limits of random outerplanar maps with independent link-weights

    Stufler, Benedikt
    The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edge-rooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us to treat subclasses such as bipartite outerplanar maps.

  6. SPDEs on narrow domains and on graphs: An asymptotic approach

    Cerrai, Sandra; Freidlin, Mark
    We introduce here a class of stochastic partial differential equations defined on a graph and we show how they are obtained as the limit of suitable stochastic partial equations defined in a narrow channel, as the width of the channel goes to zero.

  7. Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles

    Öz, Mehmet; Çağlar, Mine; Engländer, János
    We study a branching Brownian motion $Z$ in $\mathbb{R}^{d}$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide...

  8. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime

    Chen, Xia
    In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce...

  9. Point process convergence for branching random walks with regularly varying steps

    Bhattacharya, Ayan; Hazra, Rajat Subhra; Roy, Parthanil
    We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten–Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the $n$th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of (J. Stat. Phys. 143 (3) (2011) 420–446) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165–170)...

  10. Supercritical behavior of asymmetric zero-range process with sitewise disorder

    Bahadoran, C.; Mountford, T.; Ravishankar, K.; Saada, E.
    We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for one-dimensional asymmetric nearest-neighbour zero-range processes with non-homogeneous jump rates. The class of “environments” considered is close to that considered by (Stochastic Process. Appl. 90 (2000) 67–81), while our class of processes is broader. We also give in arbitrary dimension a simpler proof of the result of (In Asymptotics: Particles, Processes and Inverse Problems (2007) 108–120 Inst. Math. Statist.) with weaker assumptions.

  11. One-dependent coloring by finitary factors

    Holroyd, Alexander E.
    Holroyd and Liggett recently proved the existence of a stationary $1$-dependent $4$-coloring of the integers, the first stationary $k$-dependent $q$-coloring for any $k$ and $q$, and arguably the first natural finitely dependent process that is not a block factor of an i.i.d. process. That proof specifies a consistent family of finite-dimensional distributions, but does not yield a probabilistic construction on the whole integer line. Here we prove that the process can be expressed as a finitary factor of an i.i.d. process. The factor is described explicitly, and its coding radius obeys power-law tail bounds.

  12. Typical behavior of the harmonic measure in critical Galton–Watson trees

    Lin, Shen
    We study the typical behavior of the harmonic measure of balls in large critical Galton–Watson trees whose offspring distribution has finite variance. The harmonic measure considered here refers to the hitting distribution of height $n$ by simple random walk on a critical Galton–Watson tree conditioned to have height greater than $n$. We prove that, with high probability, the mass of the harmonic measure carried by a random vertex uniformly chosen from height $n$ is approximately equal to $n^{-\lambda}$, where the constant $\lambda>1$ does not depend on the offspring distribution. This universal constant $\lambda$ is equal to the first moment of...

  13. Any orthonormal basis in high dimension is uniformly distributed over the sphere

    Goldstein, Sheldon; Lebowitz, Joel L.; Tumulka, Roderich; Zanghî, Nino
    Let $\mathbb{X}^{d}$ be a real or complex Hilbert space of finite but large dimension $d$, let $\mathbb{S}(\mathbb{X}^{d})$ denote the unit sphere of $\mathbb{X}^{d}$, and let $u$ denote the normalized uniform measure on $\mathbb{S}(\mathbb{X}^{d})$. For a finite subset $B$ of $\mathbb{S}(\mathbb{X}^{d})$, we may test whether it is approximately uniformly distributed over the sphere by choosing a partition $A_{1},\ldots,A_{m}$ of $\mathbb{S}(\mathbb{X}^{d})$ and checking whether the fraction of points in $B$ that lie in $A_{k}$ is close to $u(A_{k})$ for each $k=1,\ldots,m$. We show that if $B$ is any orthonormal basis of $\mathbb{X}^{d}$ and $m$ is not too large, then, if we randomize...

  14. Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes

    Torrisi, Giovanni Luca
    We give a general inequality for the total variation distance between a Poisson distributed random variable and a first order stochastic integral with respect to a point process with stochastic intensity, constructed by embedding in a bivariate Poisson process. We apply this general inequality to first order stochastic integrals with respect to a class of nonlinear Hawkes processes, which is of interest in queueing theory, providing explicit bounds for the Poisson approximation, a quantitative Poisson limit theorem, confidence intervals and asymptotic estimates of the moments.

  15. A dynamical Curie–Weiss model of SOC: The Gaussian case

    Gorny, Matthias
    In this paper, we introduce a Markov process whose unique invariant distribution is the Curie–Weiss model of self-organized criticality (SOC) we designed and studied in (Ann. Probab. 44(1):444-478, 2016). In the Gaussian case, we prove rigorously that it is a dynamical model of SOC: the fluctuations of the sum $S_{n}(\cdot)$ of the process evolve in a time scale of order $\sqrt{n}$ and in a space scale of order $n^{3/4}$ and the limiting process is the solution of a “critical” stochastic differential equation.

  16. Path-dependent infinite-dimensional SDE with non-regular drift: An existence result

    Dereudre, David; Rœlly, Sylvie
    We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment. ¶ The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly...

  17. Scaling limits of coalescent processes near time zero

    Şengül, Batı
    In this paper we obtain scaling limits of $\Lambda$-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman’s coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman’s coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.

  18. Simple CLE in doubly connected domains

    Sheffield, Scott; Watson, Samuel S.; Wu, Hao
    Loop Ensemble ($\operatorname{CLE}_{\kappa}$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to $\operatorname{CLE}_{\kappa}$ for which the loops are simple, i.e. $\kappa\in(8/3,4]$. In (Ann. of Math. (2) 176 (2012) 1827–1917), simple $\operatorname{CLE}$ in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple $\operatorname{CLE}$ in the unit disc, any fixed interior point is almost surely surrounded by some loop of $\operatorname{CLE}$. The gasket of the collection of loops in $\operatorname{CLE}$, i.e. the set of points that are not surrounded by any...

  19. Decomposition of Lévy trees along their diameter

    Duquesne, Thomas; Wang, Minmin
    We study the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton–Watson trees. Lévy trees have been introduced by Le Gall & Le Jan (Ann. Probab. 26 (1998) 213–252) and they generalise Aldous’ Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter $r\in (0,\infty)$ is obtained by glueing at their respective roots two...

  20. Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise

    Fontbona, Joaquin; Panloup, Fabien
    We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved by Hairer that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can recover such convergence rates when $H>1/2$ and the inverse of...

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