Annales de l'Institut Henri Poincaré, Probabilités et Statistiques Articles (Project Euclid)
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Between Paouris concentration inequality and variance conjecture
http://projecteuclid.org/euclid.aihp/1273584125
<strong>B. Fleury</strong><p><strong>Source: </strong>Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2, 299--312.</p><p><strong>Abstract:</strong><br/>
We prove an almost isometric reverse Hölder inequality for the Euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.
</p>projecteuclid.org/euclid.aihp/1273584125_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTSupremum estimates for degenerate, quasilinear stochastic partial differential equationshttps://projecteuclid.org/euclid.aihp/1569398885<strong>Konstantinos Dareiotis</strong>, <strong>Benjamin Gess</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 3, 1765--1796.</p><p><strong>Abstract:</strong><br/>
We prove a priori estimates in $L_{\infty }$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon $ and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation.
</p>projecteuclid.org/euclid.aihp/1569398885_20190925040759Wed, 25 Sep 2019 04:07 EDTOn thin local sets of the Gaussian free fieldhttps://projecteuclid.org/euclid.aihp/1569398886<strong>Avelio Sepúlveda</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 3, 1797--1813.</p><p><strong>Abstract:</strong><br/>
We study how small a local set of the continuum Gaussian free field (GFF) in dimension $d$ has to be to ensure that this set is thin, which loosely speaking means that it captures no GFF mass on itself, in other words, that the field restricted to it is zero. We provide a criterion on the size of the local set for this to happen, and on the other hand, we show that this criterion is sharp by constructing small local sets that are not thin.
</p>projecteuclid.org/euclid.aihp/1569398886_20190925040759Wed, 25 Sep 2019 04:07 EDTRegular Dirichlet extensions of one-dimensional Brownian motionhttps://projecteuclid.org/euclid.aihp/1573203616<strong>Liping Li</strong>, <strong>Jiangang Ying</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1815--1849.</p><p><strong>Abstract:</strong><br/>
The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an $\mathcal{E}$-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in ( Ann. Probab. 45 (2017) 857–872), we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of $H^{1}_{\mathrm{e}}(\mathbb{R})$ in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.
</p>projecteuclid.org/euclid.aihp/1573203616_20191108040041Fri, 08 Nov 2019 04:00 ESTMetastability of one-dimensional, non-reversible diffusions with periodic boundary conditionshttps://projecteuclid.org/euclid.aihp/1573203617<strong>C. Landim</strong>, <strong>I. Seo</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1850--1889.</p><p><strong>Abstract:</strong><br/>
We consider small perturbations of a dynamical system on the one-dimensional torus. We derive sharp estimates for the pre-factor of the stationary state, we examine the asymptotic behavior of the solutions of the Hamilton–Jacobi equation for the pre-factor, we compute the capacities between disjoint sets, and we prove the metastable behavior of the process among the deepest wells following the martingale approach. We also present a bound for the probability that a Markov process hits a set before some fixed time in terms of the capacity of an enlarged process.
</p>projecteuclid.org/euclid.aihp/1573203617_20191108040041Fri, 08 Nov 2019 04:00 ESTOptimal survival strategy for branching Brownian motion in a Poissonian trap fieldhttps://projecteuclid.org/euclid.aihp/1573203618<strong>Mehmet Öz</strong>, <strong>János Engländer</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1890--1915.</p><p><strong>Abstract:</strong><br/>
We study a branching Brownian motion $Z$ with a generic branching law, evolving in $\mathbb{R}^{d}$, where a field of Poissonian traps is present. Each trap is a ball with constant radius. The traps are hard in the sense that the process is killed instantly once it enters the trap field. We focus on two cases of Poissonian fields, a uniform field and a radially decaying field, and consider an annealed environment. Using classical results on the convergence of the speed of branching Brownian motion, we establish precise annealed results on the population size of $Z$, given that it avoids the trap field, while staying alive up to time $t$. The results are stated so that each gives an ‘optimal survival strategy’ for $Z$. As corollaries of the results concerning the population size, we prove several other optimal survival strategies concerning the range of $Z$, and the size and position of clearings in $\mathbb{R}^{d}$. We also prove a result about the hitting time of a single trap by a branching system (Lemma 1), which may be useful in a completely generic setting too.
Inter alia , we answer some open problems raised in ( Markov Process. Related Fields 9 (2003) 363–389).
</p>projecteuclid.org/euclid.aihp/1573203618_20191108040041Fri, 08 Nov 2019 04:00 ESTAdaptive density estimation on bounded domainshttps://projecteuclid.org/euclid.aihp/1573203619<strong>Karine Bertin</strong>, <strong>Salima El Kolei</strong>, <strong>Nicolas Klutchnikoff</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1916--1947.</p><p><strong>Abstract:</strong><br/>
We study the estimation, in $\mathbb{L}_{p}$-norm, of density functions defined on $[0,1]^{d}$. We construct a new family of kernel density estimators that do not suffer from the so-called boundary bias problem and we propose a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth. We derive two estimators that satisfy oracle-type inequalities. They are also proved to be adaptive over a scale of anisotropic or isotropic Sobolev–Slobodetskii classes (which are particular cases of Besov or Sobolev classical classes). The main interest of the isotropic procedure is to obtain adaptive results without any restriction on the smoothness parameter.
</p>projecteuclid.org/euclid.aihp/1573203619_20191108040041Fri, 08 Nov 2019 04:00 ESTOn intermediate level sets of two-dimensional discrete Gaussian free fieldhttps://projecteuclid.org/euclid.aihp/1573203620<strong>Marek Biskup</strong>, <strong>Oren Louidor</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1948--1987.</p><p><strong>Abstract:</strong><br/>
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains $D\subset\mathbb{C}$ and describe the scaling limit, including local structure, of the level sets at heights growing as a $\lambda$-multiple of the height of the absolute maximum, for any $\lambda\in(0,1)$. We prove that, in the scaling limit, the scaled spatial position of a typical point $x$ sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in $D$ at parameter equal $\lambda$-times its critical value, the field value at $x$ has an exponential intensity measure and the configuration near $x$ reduced by the value at $x$ has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges to that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud ( Ann. Probab. 34 (2006) 962–986).
</p>projecteuclid.org/euclid.aihp/1573203620_20191108040041Fri, 08 Nov 2019 04:00 ESTOn the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensionshttps://projecteuclid.org/euclid.aihp/1573203621<strong>Ivan S. Yaroslavtsev</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 1988--2018.</p><p><strong>Abstract:</strong><br/>
We show that the canonical decomposition (comprising both the Meyer–Yoeurp and the Yoeurp decompositions) of a general $X$-valued local martingale is possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD Banach space if and only if for any $X$-valued local martingale $M$ there exist a continuous local martingale $M^{c}$, a purely discontinuous quasi-left continuous local martingale $M^{q}$, and a purely discontinuous local martingale $M^{a}$ with accessible jumps such that $M=M^{c}+M^{q}+M^{a}$. The corresponding weak $L^{1}$-estimates are provided. Important tools used in the proof are a new version of Gundy’s decomposition of continuous-time martingales and weak $L^{1}$-bounds for a certain class of vector-valued continuous-time martingale transforms.
</p>projecteuclid.org/euclid.aihp/1573203621_20191108040041Fri, 08 Nov 2019 04:00 ESTTransportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and couplinghttps://projecteuclid.org/euclid.aihp/1573203622<strong>Mateusz B. Majka</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2019--2057.</p><p><strong>Abstract:</strong><br/>
By using the mirror coupling for solutions of SDEs driven by pure jump Lévy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we develop a novel method of bounding Malliavin derivatives of solutions of SDEs with both jump and Gaussian noise, which involves the coupling technique and which might be of independent interest. The bounds we obtain are new even in the case of diffusions without jumps.
</p>projecteuclid.org/euclid.aihp/1573203622_20191108040041Fri, 08 Nov 2019 04:00 ESTParacontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson modelhttps://projecteuclid.org/euclid.aihp/1573203623<strong>Jörg Martin</strong>, <strong>Nicolas Perkowski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2058--2110.</p><p><strong>Abstract:</strong><br/>
We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distributions in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model.
</p>projecteuclid.org/euclid.aihp/1573203623_20191108040041Fri, 08 Nov 2019 04:00 ESTThe Circular Law for random regular digraphshttps://projecteuclid.org/euclid.aihp/1573203624<strong>Nicholas Cook</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2111--2167.</p><p><strong>Abstract:</strong><br/>
Let $\log^{C}n\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_{n}$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_{n}$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_{n}$.
</p>projecteuclid.org/euclid.aihp/1573203624_20191108040041Fri, 08 Nov 2019 04:00 ESTErratum: Qualitative properties of certain piecewise deterministic Markov processeshttps://projecteuclid.org/euclid.aihp/1573203625<strong>Michel Benaim</strong>, <strong>Stéphane Leborgne</strong>, <strong>Florent Malrieu</strong>, <strong>Pierre-André Zitt</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2168--2168.</p>projecteuclid.org/euclid.aihp/1573203625_20191108040041Fri, 08 Nov 2019 04:00 ESTLimit fluctuations for density of asymmetric simple exclusion processes with open boundarieshttps://projecteuclid.org/euclid.aihp/1573203626<strong>Włodzimierz Bryc</strong>, <strong>Yizao Wang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2169--2194.</p><p><strong>Abstract:</strong><br/>
We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier result by Derrida et al. ( J. Statist. Phys. 115 (2004) 365–382) for totally asymmetric simple exclusion process in the same phase. In the low/high density phases, the limit fluctuations are Brownian motion. Most interestingly, at the boundary of the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian meander (or a time-reversal of the latter, depending on the side of the boundary). Our proofs rely on a representation of the joint generating function of the asymmetric simple exclusion process with respect to the stationary distribution in terms of joint moments of a Markov processes, which is constructed from orthogonality measures of the Askey–Wilson polynomials.
</p>projecteuclid.org/euclid.aihp/1573203626_20191108040041Fri, 08 Nov 2019 04:00 ESTAsymptotic nonequivalence of density estimation and Gaussian white noise for small densitieshttps://projecteuclid.org/euclid.aihp/1573203627<strong>Kolyan Ray</strong>, <strong>Johannes Schmidt-Hieber</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2195--2208.</p><p><strong>Abstract:</strong><br/>
It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities are sufficiently smooth and uniformly bounded away from zero. We show that a uniform lower bound, whose size we sharply characterize, is in general necessary for asymptotic equivalence to hold.
</p>projecteuclid.org/euclid.aihp/1573203627_20191108040041Fri, 08 Nov 2019 04:00 ESTDiscretisation of regularity structureshttps://projecteuclid.org/euclid.aihp/1573203628<strong>Dirk Erhard</strong>, <strong>Martin Hairer</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2209--2248.</p><p><strong>Abstract:</strong><br/>
We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale $\varepsilon>0$ and a “black box” describing the behaviour of our discretised objects at scales below $\varepsilon$.
</p>projecteuclid.org/euclid.aihp/1573203628_20191108040041Fri, 08 Nov 2019 04:00 ESTOn time scales and quasi-stationary distributions for multitype birth-and-death processeshttps://projecteuclid.org/euclid.aihp/1573203629<strong>J.-R. Chazottes</strong>, <strong>P. Collet</strong>, <strong>S. Méléard</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2249--2294.</p><p><strong>Abstract:</strong><br/>
We consider a class of birth-and-death processes describing a population made of $d$ sub-populations of different types which interact with one another. The state space is $\mathbb{Z}^{d}_{+}$ (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter $K$ which can be thought as the order of magnitude of the total size of the population at time $0$. For any fixed finite time span, it is well-known that such processes, when renormalized by $K$, are close, in the limit $K\to+\infty$, to the solutions of a certain differential equation in $\mathbb{R}_{+}^{d}$ whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for $K$ large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time $t$ and the qsd. This bound is exponentially small in $t$, for $t\gg\log K$. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than $\log K$ and much smaller than the mean time to extinction, which is exponentially large as a function of $K$. Let us stress that we are interested in what happens for finite $K$. We obtain results much beyond what large deviation techniques could provide.
</p>projecteuclid.org/euclid.aihp/1573203629_20191108040041Fri, 08 Nov 2019 04:00 ESTReversibility of the non-backtracking random walkhttps://projecteuclid.org/euclid.aihp/1573203630<strong>Jonathan Hermon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2295--2319.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a connected graph of uniformly bounded degree. A $k$ non-backtracking random walk ($k$-NBRW) $(X_{n})_{n=0}^{\infty}$ on $G$ evolves according to the following rule: Given $(X_{n})_{n=0}^{s}$, at time $s+1$ the walk picks at random some edge which is incident to $X_{s}$ that was not crossed in the last $k$ steps and moves to its other end-point. If no such edge exists then it makes a simple random walk step. Assume that for some $R>0$ every ball of radius $R$ in $G$ contains a cycle. We show that under some “nice” random time change the $1$-NBRW becomes reversible. This is used to prove that it is recurrent iff the simple random walk is recurrent. A similar result is proved for every $k$ under stronger assumptions in general, and with no assumptions for Cayley graphs of finitely generated Abelian groups.
</p>projecteuclid.org/euclid.aihp/1573203630_20191108040041Fri, 08 Nov 2019 04:00 ESTTube estimates for diffusions under a local strong Hörmander conditionhttps://projecteuclid.org/euclid.aihp/1573203631<strong>Vlad Bally</strong>, <strong>Lucia Caramellino</strong>, <strong>Paolo Pigato</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2320--2369.</p><p><strong>Abstract:</strong><br/>
We study lower and upper bounds for the probability that a diffusion process in $\mathbb{R}^{n}$ remains in a tube around a deterministic skeleton path up to a fixed time. The diffusion coefficients $\sigma_{1},\ldots,\sigma_{d}$ may degenerate, but we assume that they satisfy a strong Hörmander condition involving the first order Lie brackets around the skeleton of interest. The tube is written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $\delta$, the diffusion process propagates with speed $\sqrt{\delta}$ in the direction of the diffusion vector fields $\sigma_{j}$ and with speed $\delta$ in the direction of $[\sigma_{i},\sigma_{j}]$. We first prove short-time (non-asymptotic) lower and upper bounds for the density of the diffusion. Then, we prove the tube estimate using a concatenation of these short-time density estimates.
</p>projecteuclid.org/euclid.aihp/1573203631_20191108040041Fri, 08 Nov 2019 04:00 ESTSticky couplings of multidimensional diffusions with different driftshttps://projecteuclid.org/euclid.aihp/1573203632<strong>Andreas Eberle</strong>, <strong>Raphael Zimmer</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2370--2394.</p><p><strong>Abstract:</strong><br/>
We present a novel approach of coupling two multi-dimensional and non-degenerate Itô processes $(X_{t})$ and $(Y_{t})$ which follow dynamics with different drifts. Our coupling is sticky in the sense that there is a stochastic process $(r_{t})$, which solves a one-dimensional stochastic differential equation with a sticky boundary behavior at zero, such that almost surely $|X_{t}-Y_{t}|\leq r_{t}$ for all $t\geq0$. The coupling is constructed as a weak limit of Markovian couplings. We provide explicit, non-asymptotic and long-time stable bounds for the probability of the event $\{X_{t}=Y_{t}\}$.
</p>projecteuclid.org/euclid.aihp/1573203632_20191108040041Fri, 08 Nov 2019 04:00 ESTOn the boundary of the zero set of super-Brownian motion and its local timehttps://projecteuclid.org/euclid.aihp/1573203633<strong>Thomas Hughes</strong>, <strong>Edwin Perkins</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 4, 2395--2422.</p><p><strong>Abstract:</strong><br/>
If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that
\[\operatorname{dim}\bigl(\partial\bigl\{x:X(t,x)>0\bigl\}\bigl)=2-2\lambda_{0}\in(0,1)\quad\text{a.s. on }\{X_{t}\neq0\},\] where $-\lambda_{0}\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein–Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins ( Ann. Probab. 45 (2017) 3481–3543) who proved the above with positive probability. To establish this result we derive some new basic properties of a boundary local time recently introduced by one of us (Hughes), and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $\lambda_{0}$ suggest that the above Hausdorff dimension is approximately $0.224$.
</p>projecteuclid.org/euclid.aihp/1573203633_20191108040041Fri, 08 Nov 2019 04:00 ESTBrownian motion in attenuated or renormalized inverse-square Poisson potentialhttps://projecteuclid.org/euclid.aihp/1580720478<strong>Peter Nelson</strong>, <strong>Renato Soares dos Santos</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 1--35.</p><p><strong>Abstract:</strong><br/>
We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^{d}$, $d\ge 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x)\approx \theta |x|^{-2}$ near the origin, where $\theta \in (0,(d-2)^{2}/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized , as introduced by Chen and Kulik in ( Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 631–660). Our main results include existence and large-time asymptotics of non-negative solutions via Feynman–Kac representation. In particular, we settle for the renormalized potential in $d=3$ the existence problem with critical parameter $\theta =1/16$, left open by Chen and Rosinski in (Chen and Rosinski (2011)).
</p>projecteuclid.org/euclid.aihp/1580720478_20200203040149Mon, 03 Feb 2020 04:01 ESTNonparametric density estimation from observations with multiplicative measurement errorshttps://projecteuclid.org/euclid.aihp/1580720482<strong>Denis Belomestny</strong>, <strong>Alexander Goldenshluger</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 36--67.</p><p><strong>Abstract:</strong><br/>
In this paper we study the problem of pointwise density estimation from observations with multiplicative measurement errors. We elucidate the main feature of this problem: the influence of the estimation point on the estimation accuracy. In particular, we show that, depending on whether this point is separated away from zero or not, there are two different regimes in terms of the rates of convergence of the minimax risk. In both regimes we develop kernel-type density estimators and prove upper bounds on their maximal risk over suitable nonparametric classes of densities. We show that the proposed estimators are rate-optimal by establishing matching lower bounds on the minimax risk. Finally we test our estimation procedures on simulated data.
</p>projecteuclid.org/euclid.aihp/1580720482_20200203040149Mon, 03 Feb 2020 04:01 ESTExponentially slow mixing in the mean-field Swendsen–Wang dynamicshttps://projecteuclid.org/euclid.aihp/1580720483<strong>Reza Gheissari</strong>, <strong>Eyal Lubetzky</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 68--86.</p><p><strong>Abstract:</strong><br/>
Swendsen–Wang dynamics for the Potts model was proposed in the late 1980’s as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum ( J. Stat. Phys. 97 (1999) 67–86) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with $q\geq 3$ colors on the complete graph on $n$ vertices at the critical point $\beta_{c}(q)$, Swendsen–Wang dynamics has $t_{\mathrm{mix}}\geq \exp (c\sqrt{n})$. Galanis et al. (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 815–828) showed that $t_{\mathrm{mix}}\geq \exp (cn^{1/3})$ throughout the critical window $(\beta_{s},\beta_{S})$ around $\beta_{c}$, and Blanca and Sinclair (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 528–543) established that $t_{\mathrm{mix}}\geq \exp (c\sqrt{n})$ in the critical window for the corresponding mean-field FK model, which implied the same bound for Swendsen–Wang via known comparison estimates. In both cases, an upper bound of $t_{\mathrm{mix}}\leq \exp (c'n)$ was known. Here we show that the mixing time is truly exponential in $n$: namely, $t_{\mathrm{mix}}\geq \exp (cn)$ for Swendsen–Wang dynamics when $q\geq 3$ and $\beta \in (\beta_{s},\beta_{S})$, and the same bound holds for the related MCMC samplers for the mean-field FK model when $q>2$.
</p>projecteuclid.org/euclid.aihp/1580720483_20200203040149Mon, 03 Feb 2020 04:01 ESTOn the thin-shell conjecture for the Schatten classeshttps://projecteuclid.org/euclid.aihp/1580720484<strong>Jordan Radke</strong>, <strong>Beatrice-Helen Vritsiou</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 87--119.</p><p><strong>Abstract:</strong><br/>
We study the thin-shell conjecture for the Schatten classes. In particular, we establish the conjecture for the operator norm, and we also improve on the best known bound for the Schatten classes, due to Barthe and Cordero-Erausquin ( Proc. Lond. Math. Soc. 106 (2013) 33–64) or Lee and Vempala (2017), for a few more cases. We also show that a necessary condition for the conjecture to be true for any of the Schatten classes is a rather strong negative correlation property: as a consequence of this we obtain the validity of this negative correlation property for all the cases for which we already know the conjecture is true (as for example for the operator norm), but moreover also for all the cases for which we can get a better estimate than the one in ( Proc. Lond. Math. Soc. 106 (2013) 33–64) or (Lee and Vempala (2017)). For the proofs, our starting point is techniques that were employed for the Schatten classes in ( Math. Ann. 312 (1998) 773–783) and ( Ann. Inst. Henri Poincaré Probab. Stat. 43 (2007) 87–99) with regard to other problems.
</p>projecteuclid.org/euclid.aihp/1580720484_20200203040149Mon, 03 Feb 2020 04:01 ESTSpectral statistics of sparse Erdős–Rényi graph Laplacianshttps://projecteuclid.org/euclid.aihp/1580720485<strong>Jiaoyang Huang</strong>, <strong>Benjamin Landon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 120--154.</p><p><strong>Abstract:</strong><br/>
We consider the bulk eigenvalue statistics of Laplacian matrices of large Erdős–Rényi random graphs in the regime $p\geq N^{\delta }/N$ for any fixed $\delta >0$. We prove a local law down to the optimal scale $\eta \gtrsim N^{-1}$ which implies that the eigenvectors are delocalized. We consider the local eigenvalue statistics and prove that both the gap statistics and averaged correlation functions coincide with the GOE in the bulk.
</p>projecteuclid.org/euclid.aihp/1580720485_20200203040149Mon, 03 Feb 2020 04:01 ESTMarkovian integral equationshttps://projecteuclid.org/euclid.aihp/1580720486<strong>Alexander Kalinin</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 155--174.</p><p><strong>Abstract:</strong><br/>
We analyze multidimensional Markovian integral equations that are formulated with a progressive time-inhomogeneous Markov process that has Borel measurable transition probabilities. In the case of a path process of a path-dependent diffusion, the solutions to these integral equations lead to the concept of mild solutions to path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence and non-extendibility of solutions among a certain class of maps. By requiring the Feller continuity of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman–Kac formula and a one-dimensional global existence and uniqueness result.
</p>projecteuclid.org/euclid.aihp/1580720486_20200203040149Mon, 03 Feb 2020 04:01 ESTErgodicity of stochastic differential equations with jumps and singular coefficientshttps://projecteuclid.org/euclid.aihp/1580720487<strong>Longjie Xie</strong>, <strong>Xicheng Zhang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 175--229.</p><p><strong>Abstract:</strong><br/>
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drifts. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov’s a priori estimates for SDEs.
</p>projecteuclid.org/euclid.aihp/1580720487_20200203040149Mon, 03 Feb 2020 04:01 ESTStatistical limits of spiked tensor modelshttps://projecteuclid.org/euclid.aihp/1580720488<strong>Amelia Perry</strong>, <strong>Alexander S. Wein</strong>, <strong>Afonso S. Bandeira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 230--264.</p><p><strong>Abstract:</strong><br/>
We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. $\pm1$ entries, and (iii) a sparse vector where a constant fraction $\rho$ of entries are i.i.d. $\pm1$ and the rest are zero. For each of these cases, our upper and lower bounds match up to a $1+o(1)$ factor as the order $d$ of the tensor becomes large. For sparse signals (iii), our bounds are also asymptotically tight in the sparse limit $\rho\to0$ for any fixed $d$ (including the $d=2$ case of sparse PCA). Our upper bounds for (i) demonstrate a phenomenon reminiscent of the work of Baik, Ben Arous and Péché: an ‘eigenvalue’ of a perturbed tensor emerges from the bulk at a strictly lower signal-to-noise ratio than when the perturbation itself exceeds the bulk; we quantify the size of this effect. We also provide some general results for larger classes of priors. In particular, the large $d$ asymptotics of the threshold location differs between problems with discrete priors versus continuous priors. Finally, for priors (i) and (ii) we carry out the replica prediction from statistical physics, which is conjectured to give the exact information-theoretic threshold for any fixed $d$.
Of independent interest, we introduce a new improvement to the second moment method for contiguity, on which our lower bounds are based. Our technique conditions away from rare ‘bad’ events that depend on interactions between the signal and noise. This enables us to close $\sqrt{2}$-factor gaps present in several previous works.
</p>projecteuclid.org/euclid.aihp/1580720488_20200203040149Mon, 03 Feb 2020 04:01 ESTOn laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivityhttps://projecteuclid.org/euclid.aihp/1580720489<strong>Matthieu Jonckheere</strong>, <strong>Santiago Saglietti</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 265--295.</p><p><strong>Abstract:</strong><br/>
We give necessary and sufficient conditions for laws of large numbers to hold in $L^{2}$ for the empirical measure of a large class of branching Markov processes, including $\lambda $-positive systems but also some $\lambda $-transient ones, such as the branching Brownian motion with drift and absorption at $0$. This is a significant improvement over previous results on this matter, which had only dealt so far with $\lambda $-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.
</p>projecteuclid.org/euclid.aihp/1580720489_20200203040149Mon, 03 Feb 2020 04:01 ESTSome properties of the free stable distributionshttps://projecteuclid.org/euclid.aihp/1580720490<strong>Takahiro Hasebe</strong>, <strong>Thomas Simon</strong>, <strong>Min Wang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 296--325.</p><p><strong>Abstract:</strong><br/>
We investigate certain analytical properties of the free $\alpha $-stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha \le 1$ and that they belong to the extended Thorin class when $\alpha \le 3/4$. The Lévy measure is explicitly computed for $\alpha =1$, showing that free 1-stable distributions are not in the Thorin class except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\alpha >1$. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped, that is their successive derivatives vanish exactly once on their support. We also derive several fine properties of spectrally one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, and several intrinsic features of whale-shaped functions.
</p>projecteuclid.org/euclid.aihp/1580720490_20200203040149Mon, 03 Feb 2020 04:01 ESTExistence of densities for the dynamic $\Phi^{4}_{3}$ modelhttps://projecteuclid.org/euclid.aihp/1580720491<strong>Paul Gassiat</strong>, <strong>Cyril Labbé</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 326--373.</p><p><strong>Abstract:</strong><br/>
We apply Malliavin calculus to the $\Phi^{4}_{3}$ equation on the torus and prove existence of densities for the solution of the equation evaluated at regular enough test functions. We work in the framework of regularity structures and rely on Besov-type spaces of modelled distributions in order to prove Malliavin differentiability of the solution. Our result applies to a large family of Gaussian space–time noises including white noise, in particular the noise may be degenerate as long as it is sufficiently rough on small scales.
</p>projecteuclid.org/euclid.aihp/1580720491_20200203040149Mon, 03 Feb 2020 04:01 ESTThe asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonicshttps://projecteuclid.org/euclid.aihp/1580720492<strong>Domenico Marinucci</strong>, <strong>Maurizia Rossi</strong>, <strong>Igor Wigman</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 374--390.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell\rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the “sample trispectrum”, i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.
</p>projecteuclid.org/euclid.aihp/1580720492_20200203040149Mon, 03 Feb 2020 04:01 ESTInternal diffusion-limited aggregation with uniform starting pointshttps://projecteuclid.org/euclid.aihp/1580720493<strong>Itai Benjamini</strong>, <strong>Hugo Duminil-Copin</strong>, <strong>Gady Kozma</strong>, <strong>Cyrille Lucas</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 391--404.</p><p><strong>Abstract:</strong><br/>
We study internal diffusion-limited aggregation with uniform starting points on $\mathbb{Z}^{d}$. In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ball.
</p>projecteuclid.org/euclid.aihp/1580720493_20200203040149Mon, 03 Feb 2020 04:01 ESTEmpirical risk minimization as parameter choice rule for general linear regularization methodshttps://projecteuclid.org/euclid.aihp/1580720494<strong>Housen Li</strong>, <strong>Frank Werner</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 405--427.</p><p><strong>Abstract:</strong><br/>
We consider the statistical inverse problem to recover $f$ from noisy measurements $Y=Tf+\sigma \xi $ where $\xi $ is Gaussian white noise and $T$ a compact operator between Hilbert spaces. Considering general reconstruction methods of the form $\hat{f}_{\alpha }=q_{\alpha }(T^{*}T)T^{*}Y$ with an ordered filter $q_{\alpha }$, we investigate the choice of the regularization parameter $\alpha $ by minimizing an unbiased estimate of the predictive risk $\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. The corresponding parameter $\alpha_{\mathrm{pred}}$ and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk $\mathbb{E}[\Vert f-\hat{f}_{\alpha_{\mathrm{pred}}}\Vert^{2}]$ with the oracle prediction risk $\inf_{\alpha >0}\mathbb{E}[\Vert Tf-T\hat{f}_{\alpha }\Vert^{2}]$. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense.
Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations.
</p>projecteuclid.org/euclid.aihp/1580720494_20200203040149Mon, 03 Feb 2020 04:01 ESTNonconventional moderate deviations theorems and exponential concentration inequalitieshttps://projecteuclid.org/euclid.aihp/1580720495<strong>Yeor Hafouta</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 428--448.</p><p><strong>Abstract:</strong><br/>
We obtain moderate deviations theorems and exponential (Bernstein type) concentration inequalities for “nonconventional” sums of the form $S_{N}=\sum_{n=1}^{N}(F(\xi_{q_{1}(n)},\xi_{q_{2}(n)},\ldots,\xi_{q_{\ell }(n)})-\bar{F})$, where most of the time we consider $q_{i}(n)=in$, but our results also hold true for more general $q_{i}(n)$’s such as polynomials. Here $\xi_{n}$, $n\geq 0$ is a sufficiently fast mixing vector process with some stationarity conditions, $F$ is a function satisfying certain regularity conditions and $\bar{F}$ is a certain centralizing constant. When $\xi_{n}$, $n\geq 0$ are independent and identically distributed a large deviations theorem was obtained in ( Probab. Theory Related Fields 158 (2014) 197–224) and one of the purposes of this paper is to obtain related results in the (weakly) dependent case. Several normal approximation type results will also be derived. In particular, two more proofs of the nonconventional central limit theorem are given and a Rosenthal type inequality is obtained. Our results hold true, for instance, when $\xi_{n}=(T^{n}f_{i})_{i=1}^{\wp }$ where $T$ is a topologically mixing subshift of finite type, a Gibbs–Markov map, a hyperbolic diffeomorphism, a Young tower or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $\xi_{n}$, $n\geq 0$ forms a stationary and (stretched) exponentially fast $\phi $-mixing sequence, which, for instance, holds true when $\xi_{n}=(f_{i}(\Upsilon_{n}))_{i=1}^{\wp }$ where $\Upsilon_{n}$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.
</p>projecteuclid.org/euclid.aihp/1580720495_20200203040149Mon, 03 Feb 2020 04:01 ESTRenewal theorems and mixing for non Markov flows with infinite measurehttps://projecteuclid.org/euclid.aihp/1580720496<strong>Ian Melbourne</strong>, <strong>Dalia Terhesiu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 449--476.</p><p><strong>Abstract:</strong><br/>
We obtain results on mixing for a large class of (not necessarily Markov) infinite measure semiflows and flows. Erickson proved, amongst other things, a strong renewal theorem in the corresponding i.i.d. setting. Using operator renewal theory, we extend Erickson’s methods to the deterministic (i.e. non-i.i.d.) continuous time setting and obtain results on mixing as a consequence.
Our results apply to intermittent semiflows and flows of Pomeau–Manneville type (both Markov and nonMarkov), and to semiflows and flows over Collet–Eckmann maps with nonintegrable roof function.
</p>projecteuclid.org/euclid.aihp/1580720496_20200203040149Mon, 03 Feb 2020 04:01 ESTOn a non-linear 2D fractional wave equationhttps://projecteuclid.org/euclid.aihp/1580720497<strong>Aurélien Deya</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 477--501.</p><p><strong>Abstract:</strong><br/>
We investigate the following non-linear stochastic wave equation model: \begin{equation*}\begin{cases}\partial^{2}_{t}u-\Delta u=u^{2}+\dot{B},\quad t\in[0,T],x\in\mathbb{R}^{2},\\u(0,\cdot)=\phi_{0},\qquad \partial_{t}u(0,\cdot)=\phi_{1},\end{cases}\end{equation*} where $\phi_{0},\phi_{1}$ are deterministic initial conditions in an appropriate Sobolev space and $\dot{B}$ stands for a space–time fractional noise. In this two-dimensional situation, we develop a strategy based on a third-order expansion of the equation, which, combined with a Wick-renormalization procedure, allows us to extend the results of Deya (2019) to a rougher noise.
We also point out the limits of this specific strategy when considering a highly rough noise.
</p>projecteuclid.org/euclid.aihp/1580720497_20200203040149Mon, 03 Feb 2020 04:01 ESTScaling limits of discrete snakes with stable branchinghttps://projecteuclid.org/euclid.aihp/1580720498<strong>Cyril Marzouk</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 502--523.</p><p><strong>Abstract:</strong><br/>
We consider so-called discrete snakes obtained from size-conditioned critical Bienaymé–Galton–Watson trees by assigning to each node a random spatial position in such a way that the increments along each edge are i.i.d. When the offspring distribution belongs to the domain of attraction of a stable law with index $\alpha\in(1,2]$, we give a necessary and sufficient condition on the tail distribution of the spatial increments for this spatial tree to converge, in a functional sense, towards the Brownian snake driven by the $\alpha$-stable Lévy tree. We also study the case of heavier tails, and apply our result to study the number of inversions of a uniformly random permutation indexed by the tree.
</p>projecteuclid.org/euclid.aihp/1580720498_20200203040149Mon, 03 Feb 2020 04:01 ESTQuasi-static large deviationshttps://projecteuclid.org/euclid.aihp/1580720499<strong>Anna De Masi</strong>, <strong>Stefano Olla</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 524--542.</p><p><strong>Abstract:</strong><br/>
We consider the symmetric simple exclusion with open boundaries that are in contact with particle reservoirs at different densities. The reservoir densities changes at a slower time scale with respect to the natural time scale the system reaches the stationary state.This gives rise to the quasi static hydrodynamic limit proven in ( Journal of Statistical Physics 161 (5) (2015) 1037–1058). We study here the large deviations with respect to this limit for the particle density field and the total current. We identify explicitely the large deviation functional and prove that it satisfies a fluctuation relation.
</p>projecteuclid.org/euclid.aihp/1580720499_20200203040149Mon, 03 Feb 2020 04:01 ESTCouplings in $L^{p}$ distance of two Brownian motions and their Lévy areahttps://projecteuclid.org/euclid.aihp/1580720500<strong>Michel Bonnefont</strong>, <strong>Nicolas Juillet</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 543--565.</p><p><strong>Abstract:</strong><br/>
We study co-adapted couplings of (canonical hypoelliptic) diffusions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brownian motions and are the joint laws of a planar Brownian motion with its Lévy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time $t\geq 0$. Actually, we prove the stronger result that they can not stay bounded in $L^{p}$ for $p\geq 2$.
We also prove two positive results. We first study the coupling by reflection and show that it stays bounded in $L^{p}$ for $0\leq p<1$. Secondly, we construct an explicit static (and in particular non co-adapted) coupling between the laws of two Brownian motions, which provides $L^{1}$-Wasserstein control uniformly in time.
Finally, we explain how the results generalise to the Heisenberg groups of higher dimension.
</p>projecteuclid.org/euclid.aihp/1580720500_20200203040149Mon, 03 Feb 2020 04:01 ESTBounds on the Poincaré constant for convolution measureshttps://projecteuclid.org/euclid.aihp/1580720501<strong>Thomas A. Courtade</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 566--579.</p><p><strong>Abstract:</strong><br/>
We establish a Shearer-type inequality for the Poincaré constant, showing that the Poincaré constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincaré constants corresponding to convolutions of subsets of measures. This implies, for example, that the sequence of Poincaré constants corresponding to successive convolutions in the central limit theorem is non-increasing. We also establish a dimension-free stability estimate for subadditivity of the Poincaré constant on convolutions which uniformly improves an earlier one-dimensional estimate of a similar nature by Johnson ( Teor. Veroyatn. Primen. 48 (2003) 615–620). As a byproduct of our arguments, we find that the various monotone properties of entropy, Fisher information and the Poincaré constant on convolutions have a common, simple root in Shearer’s inequality.
</p>projecteuclid.org/euclid.aihp/1580720501_20200203040149Mon, 03 Feb 2020 04:01 ESTA growth-fragmentation model related to Ornstein–Uhlenbeck type processeshttps://projecteuclid.org/euclid.aihp/1580720502<strong>Quan Shi</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 580--611.</p><p><strong>Abstract:</strong><br/>
Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which have focused mainly on the self-similar case, we introduce a new type of growth-fragmentation which is closely related to Lévy driven Ornstein–Uhlenbeck type processes. Our model can be viewed as a generalization of compensated fragmentation processes introduced by Bertoin, or the stochastic counterpart of a family of growth-fragmentation equations. We establish a convergence criterion for a sequence of such growth-fragmentations. We also prove that, under certain conditions, this system fulfills a law of large numbers.
</p>projecteuclid.org/euclid.aihp/1580720502_20200203040149Mon, 03 Feb 2020 04:01 ESTGradient bounds for Kolmogorov type diffusionshttps://projecteuclid.org/euclid.aihp/1580720503<strong>Fabrice Baudoin</strong>, <strong>Maria Gordina</strong>, <strong>Phanuel Mariano</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 612--636.</p><p><strong>Abstract:</strong><br/>
We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $\Gamma $-calculus techniques. The advantages and drawbacks of each of these methods are discussed.
</p>projecteuclid.org/euclid.aihp/1580720503_20200203040149Mon, 03 Feb 2020 04:01 ESTA central limit theorem for Fleming–Viot particle systemshttps://projecteuclid.org/euclid.aihp/1580720504<strong>Frédéric Cérou</strong>, <strong>Bernard Delyon</strong>, <strong>Arnaud Guyader</strong>, <strong>Mathias Rousset</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 637--666.</p><p><strong>Abstract:</strong><br/>
Fleming–Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously from the state of another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, the key suppositions are that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. In particular, this includes the case of elliptic diffusions with hard killing.
</p>projecteuclid.org/euclid.aihp/1580720504_20200203040149Mon, 03 Feb 2020 04:01 ESTHydrodynamic limit for a facilitated exclusion processhttps://projecteuclid.org/euclid.aihp/1580720505<strong>Oriane Blondel</strong>, <strong>Clément Erignoux</strong>, <strong>Makiko Sasada</strong>, <strong>Marielle Simon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 667--714.</p><p><strong>Abstract:</strong><br/>
We study the hydrodynamic limit for a periodic $1$-dimensional exclusion process with a dynamical constraint, which prevents a particle at site $x$ from jumping to site $x\pm 1$ unless site $x\mp 1$ is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than $\frac{1}{2}$, or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density $\frac{1}{2}$, the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.
</p>projecteuclid.org/euclid.aihp/1580720505_20200203040149Mon, 03 Feb 2020 04:01 ESTThe existence phase transition for scale invariant Poisson random fractal modelshttps://projecteuclid.org/euclid.aihp/1580720506<strong>Erik I. Broman</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 715--733.</p><p><strong>Abstract:</strong><br/>
In this paper we study the existence phase transition of scale invariant random fractal models. We determine the exact value of the critical point of this phase transition for all models satisfying some weak assumptions. In addition, we show that for a large subclass, the fractal model is in the empty phase at the critical point. This subclass of models includes the scale invariant Poisson Boolean model and the Brownian loop soup. In contrast to earlier results in the literature, we do not need to restrict our attention to random fractal models generated by open sets.
</p>projecteuclid.org/euclid.aihp/1580720506_20200203040149Mon, 03 Feb 2020 04:01 ESTRecurrence of Markov chain traceshttps://projecteuclid.org/euclid.aihp/1580720507<strong>Itai Benjamini</strong>, <strong>Jonathan Hermon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 734--759.</p><p><strong>Abstract:</strong><br/>
It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the square grid $\mathbb{Z}^{2}$. In particular, the $d$-dimensional grid $\mathbb{Z}^{d}$ admits such a Markov chain only when $d=2$. For $d=2$ we present a relevant example due to Gady Kozma, while the general statement for transient graphs is obtained by proving that for every transient irreducible Markov chain on a countable state space which admits a stationary measure, its trace is almost surely recurrent for simple random walk. The case that the Markov chain is reversible is due to Gurel-Gurevich, Lyons and the first named author (2007). We exploit recent results in potential theory of non-reversible Markov chains in order to extend their result to the non-reversible setup.
</p>projecteuclid.org/euclid.aihp/1580720507_20200203040149Mon, 03 Feb 2020 04:01 ESTErrata for Perturbation by non-local operatorshttps://projecteuclid.org/euclid.aihp/1580720508<strong>Zhen-Qing Chen</strong>, <strong>Jie-Ming Wang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 1, 760--763.</p>projecteuclid.org/euclid.aihp/1580720508_20200203040149Mon, 03 Feb 2020 04:01 ESTPath-space moderate deviations for a Curie–Weiss model of self-organized criticalityhttps://projecteuclid.org/euclid.aihp/1584345618<strong>Francesca Collet</strong>, <strong>Matthias Gorny</strong>, <strong>Richard C. Kraaij</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 765--781.</p><p><strong>Abstract:</strong><br/>
The dynamical Curie–Weiss model of self-organized criticality (SOC) was introduced in ( Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678) and it is derived from the classical generalized Curie–Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie–Weiss model of SOC ( Ann. Probab. 44 (2016) 444–478) as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton–Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.
</p>projecteuclid.org/euclid.aihp/1584345618_20200316040027Mon, 16 Mar 2020 04:00 EDTDivergence of shape fluctuation for general distributions in first-passage percolationhttps://projecteuclid.org/euclid.aihp/1584345619<strong>Shuta Nakajima</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 782--791.</p><p><strong>Abstract:</strong><br/>
We study the shape fluctuation in the first-passage percolation on $\mathbb{Z}^{d}$. It is known that it diverges when the distribution obeys Bernoulli in Zhang ( Probab. Theory. Related. Fields. 136 (2006) 298–320). In this paper, we extend the result to general distributions.
</p>projecteuclid.org/euclid.aihp/1584345619_20200316040027Mon, 16 Mar 2020 04:00 EDTParabolic Anderson model with a fractional Gaussian noise that is rough in timehttps://projecteuclid.org/euclid.aihp/1584345620<strong>Xia Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 792--825.</p><p><strong>Abstract:</strong><br/>
This paper concerns the parabolic Anderson equation \begin{equation*}\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\,\partial x_{1}\cdots \,\partial x_{d}} \end{equation*} generated by a $(d+1)$-dimensional fractional noise with the Hurst parameter $\mathbf{H}=(H_{0},H_{1},\ldots ,H_{d})$ with special interest in the setting that some of $H_{0},\ldots ,H_{d}$ are less than half. In the recent work ( Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 941–976), the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when $H_{0}<1/2$ with the concern on solvability, Feynman–Kac’s moment formula and intermittency of the system.
</p>projecteuclid.org/euclid.aihp/1584345620_20200316040027Mon, 16 Mar 2020 04:00 EDTStable matchings in high dimensions via the Poisson-weighted infinite treehttps://projecteuclid.org/euclid.aihp/1584345621<strong>Alexander E. Holroyd</strong>, <strong>James B. Martin</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 826--846.</p><p><strong>Abstract:</strong><br/>
We consider the stable matching of two independent Poisson processes in $\mathbb{R}^{d}$ under an asymmetric color restriction. Blue points can only match to red points, while red points can match to points of either color. It is unknown whether there exists a choice of intensities of the red and blue processes under which all points are matched. We prove that for any fixed intensities, there are unmatched blue points in sufficiently high dimension. Indeed, if the ratio of red to blue intensities is $\rho $ then the intensity of unmatched blue points converges to $e^{-\rho }/(1+\rho )$ as $d\to \infty $. We also establish analogous results for certain multi-color variants. Our proof uses stable matching on the Poisson-weighted infinite tree (PWIT), which can be analyzed via differential equations. The PWIT has been used in many settings as a scaling limit for models involving complete graphs with independent edge weights, but as far as we are aware, this is the first presentation of a rigorous application to high-dimensional Euclidean space. Finally, we analyze the asymmetric matching problem under a hierarchical metric, and show that there are unmatched points for all intensities.
</p>projecteuclid.org/euclid.aihp/1584345621_20200316040027Mon, 16 Mar 2020 04:00 EDTInfinite rate symbiotic branching on the real line: The tired frogs modelhttps://projecteuclid.org/euclid.aihp/1584345622<strong>Achim Klenke</strong>, <strong>Leonid Mytnik</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 847--883.</p><p><strong>Abstract:</strong><br/>
Consider a population of infinitesimally small frogs on the real line. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. This frog model is a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types. We construct this frog model as the limit of approximating processes and compute the structure of jumps. We show that our frog model can be described by a stochastic partial differential equation on the real line with a jump type noise.
</p>projecteuclid.org/euclid.aihp/1584345622_20200316040027Mon, 16 Mar 2020 04:00 EDTA perturbation analysis of stochastic matrix Riccati diffusionshttps://projecteuclid.org/euclid.aihp/1584345623<strong>Adrian N. Bishop</strong>, <strong>Pierre Del Moral</strong>, <strong>Angèle Niclas</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 884--916.</p><p><strong>Abstract:</strong><br/>
Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman–Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.
</p>projecteuclid.org/euclid.aihp/1584345623_20200316040027Mon, 16 Mar 2020 04:00 EDTIndistinguishability of collections of trees in the uniform spanning foresthttps://projecteuclid.org/euclid.aihp/1584345624<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 917--927.</p><p><strong>Abstract:</strong><br/>
We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^{d}$: Suppose that $\mathscr{A}$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. Then either every $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.
</p>projecteuclid.org/euclid.aihp/1584345624_20200316040027Mon, 16 Mar 2020 04:00 EDTSanov-type large deviations in Schatten classeshttps://projecteuclid.org/euclid.aihp/1584345625<strong>Zakhar Kabluchko</strong>, <strong>Joscha Prochno</strong>, <strong>Christoph Thäle</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 928--953.</p><p><strong>Abstract:</strong><br/>
Denote by $\lambda_{1}(A),\ldots ,\lambda_{n}(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_{n}$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to the classical $\ell_{p}^{n}$-ball, defined as the set of all self-adjoint $(n\times n)$-matrices satisfying $\sum_{k=1}^{n}|\lambda_{k}(A)|^{p} \leq 1$. We prove a large deviations principle for the (random) spectral measure of the matrix $n^{1/p}Z_{n}$. As a consequence, we obtain that the spectral measure of $n^{1/p}Z_{n}$ converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as $n\to \infty $. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.
</p>projecteuclid.org/euclid.aihp/1584345625_20200316040027Mon, 16 Mar 2020 04:00 EDTA Central Limit Theorem for Wasserstein type distances between two distinct univariate distributionshttps://projecteuclid.org/euclid.aihp/1584345626<strong>Philippe Berthet</strong>, <strong>Jean-Claude Fort</strong>, <strong>Thierry Klein</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 954--982.</p><p><strong>Abstract:</strong><br/>
In this article we study the natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions $F$ and $G$ on $\mathbb{R}$. The estimator is based on the order statistics of a sample having marginals $F$, $G$ and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order $p>1$ and compatible classical probability distributions.
</p>projecteuclid.org/euclid.aihp/1584345626_20200316040027Mon, 16 Mar 2020 04:00 EDTOn the mixing time of the Diaconis–Gangolli random walk on contingency tables over $\mathbb{Z}/q\mathbb{Z}$https://projecteuclid.org/euclid.aihp/1584345627<strong>Evita Nestoridi</strong>, <strong>Oanh Nguyen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 983--1001.</p><p><strong>Abstract:</strong><br/>
The Diaconis–Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis–Gangolli random walk restricted on $n\times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk exhibits cutoff at $\frac{n^{2}}{4(1-\cos{\frac{2 \pi}{q}})}\log n$, when $\log q=o(\frac{\sqrt{\log n}}{ \log\log n})$.
</p>projecteuclid.org/euclid.aihp/1584345627_20200316040027Mon, 16 Mar 2020 04:00 EDTWeak uniqueness and density estimates for SDEs with coefficients depending on some path-functionalshttps://projecteuclid.org/euclid.aihp/1584345628<strong>Noufel Frikha</strong>, <strong>Libo Li</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1002--1040.</p><p><strong>Abstract:</strong><br/>
In this article, we develop a methodology to prove weak uniqueness for stochastic differential equations with coefficients depending on some path-functionals of the process. As an extension of the technique developed by Bass and Perkins (In From Probability to Geometry (I): Volume in Honor of the 60th Birthday of Jean-Michel Bismut (2009) 47–53) in the standard diffusion case, the proposed methodology allows one to deal with process whose probability laws is singular with respect to the Lebesgue measure. To illustrate our methodology, we prove weak existence and uniqueness in the two following examples: a diffusion process with coefficients depending on its running local time and a diffusion process with coefficients depending on its running maximum. In each example, we also prove the existence of the associated transition density and establish some Gaussian upper-estimates.
</p>projecteuclid.org/euclid.aihp/1584345628_20200316040027Mon, 16 Mar 2020 04:00 EDTOn a toy network of neurons interacting through their dendriteshttps://projecteuclid.org/euclid.aihp/1584345629<strong>Nicolas Fournier</strong>, <strong>Etienne Tanré</strong>, <strong>Romain Veltz</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1041--1071.</p><p><strong>Abstract:</strong><br/>
Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{\mathrm{min}}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_{n}$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{\min },\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n,N\to \infty $ with $w_{n}\simeq N^{-1/2}$. We make use of some recent versions of the results of Deuschel and Zeitouni ( Ann. Probab. 23 (1995) 852–878) concerning the size of the longest increasing subsequence of an i.i.d. collection of points in the plan. We also study, in a very particular case, a slightly different model where the neurons spike when their potential reach some maximum value $v_{\mathrm{max}}$, and find an explicit formula for the (heuristic) mean-field limit.
</p>projecteuclid.org/euclid.aihp/1584345629_20200316040027Mon, 16 Mar 2020 04:00 EDTAsymptotics of free fermions in a quadratic well at finite temperature and the Moshe–Neuberger–Shapiro random matrix modelhttps://projecteuclid.org/euclid.aihp/1584345630<strong>Karl Liechty</strong>, <strong>Dong Wang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1072--1098.</p><p><strong>Abstract:</strong><br/>
We derive the local statistics of the canonical ensemble of free fermions in a quadratic potential well at finite temperature, as the particle number approaches infinity. This free fermion model is equivalent to a random matrix model proposed by Moshe, Neuberger and Shapiro. Limiting behaviors obtained before for the grand canonical ensemble are observed in the canonical ensemble: We have at the edge the phase transition from the Tracy–Widom distribution to the Gumbel distribution via the Kardar–Parisi–Zhang (KPZ) crossover distribution, and in the bulk the phase transition from the sine point process to the Poisson point process.
</p>projecteuclid.org/euclid.aihp/1584345630_20200316040027Mon, 16 Mar 2020 04:00 EDTNon-equilibrium fluctuations for the SSEP with a slow bondhttps://projecteuclid.org/euclid.aihp/1584345631<strong>D. Erhard</strong>, <strong>T. Franco</strong>, <strong>P. Gonçalves</strong>, <strong>A. Neumann</strong>, <strong>M. Tavares</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1099--1128.</p><p><strong>Abstract:</strong><br/>
We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of [ Stochastic Process. Appl. 123 (2013) 4156–4185; Stochastic Process. Appl. 126 (2016) 3235–3242], which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel’s principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.
</p>projecteuclid.org/euclid.aihp/1584345631_20200316040027Mon, 16 Mar 2020 04:00 EDTInfinite geodesics in hyperbolic random triangulationshttps://projecteuclid.org/euclid.aihp/1584345632<strong>Thomas Budzinski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1129--1161.</p><p><strong>Abstract:</strong><br/>
We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_{\lambda }$ introduced in ( Probab. Theory Related Fields 165 (2016) 509–540). We prove that these geodesics form a supercritical Galton–Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_{\lambda }$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limit arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper (Budzinski (2018)), we obtain new hyperbolicity properties of $\mathbb{T}_{\lambda }$: they satisfy a weaker form of Gromov-hyperbolicity and admit bi-infinite geodesics.
</p>projecteuclid.org/euclid.aihp/1584345632_20200316040027Mon, 16 Mar 2020 04:00 EDTOn the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficienthttps://projecteuclid.org/euclid.aihp/1584345633<strong>Thomas Müller-Gronbach</strong>, <strong>Larisa Yaroslavtseva</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1162--1178.</p><p><strong>Abstract:</strong><br/>
Recently a lot of effort has been invested to analyze the $L_{p}$-error of the Euler–Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_{p}$-error rate of at least $1/(2p)$ – has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler–Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2$ for all $p\in [1,\infty )$ as in the case of SDEs with Lipschitz coefficients. The proof of this result is based on a detailed analysis of appropriate occupation times for the Euler–Maruyama scheme.
</p>projecteuclid.org/euclid.aihp/1584345633_20200316040027Mon, 16 Mar 2020 04:00 EDTAbsence of percolation for Poisson outdegree-one graphshttps://projecteuclid.org/euclid.aihp/1584345634<strong>David Coupier</strong>, <strong>David Dereudre</strong>, <strong>Simon Le Stum</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1179--1202.</p><p><strong>Abstract:</strong><br/>
A Poisson outdegree-one graph is an oriented graph based on a Poisson point process such that each vertex has only one outgoing edge. The paper focuses on the absence of percolation for such graphs. Our main result is based on two assumptions. The Shield assumption ensures that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch of the graph merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. Several models satisfy these general assumptions and inherit in consequence the absence of percolation. In particular, we solve in Theorem 3.1 a conjecture by Daley et al. on the absence of percolation for the line-segment model (Conjecture 7.1 of ( Probab. Math. Statist. 36 (2016) 221–246), discussed in ( Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 127–145) as well). In this planar model, a segment is growing from any point of the Poisson process and stops its growth whenever it hits another segment. The random directions are picked independently and uniformly on the unit sphere. Another model of geometric navigation is presented and also fulfills the Shield and Loop assumptions.
</p>projecteuclid.org/euclid.aihp/1584345634_20200316040027Mon, 16 Mar 2020 04:00 EDTTrees within trees II: Nested fragmentationshttps://projecteuclid.org/euclid.aihp/1584345635<strong>Jean-Jil Duchamps</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1203--1229.</p><p><strong>Abstract:</strong><br/>
Similarly as in ( Electron. J. Probab. 23 (2018)) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $(\zeta,\xi)$ where $\zeta$ is finer than $\xi$. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified – one being specific to the nested setting and relating to a bivariate paintbox process.
</p>projecteuclid.org/euclid.aihp/1584345635_20200316040027Mon, 16 Mar 2020 04:00 EDTStochastic Hölder continuity of random fields governed by a system of stochastic PDEshttps://projecteuclid.org/euclid.aihp/1584345636<strong>Kai Du</strong>, <strong>Jiakun Liu</strong>, <strong>Fu Zhang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1230--1250.</p><p><strong>Abstract:</strong><br/>
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in $L^{p}$-space of random variables. A modified stochastic parabolicity condition involving $p$ is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.
</p>projecteuclid.org/euclid.aihp/1584345636_20200316040027Mon, 16 Mar 2020 04:00 EDTHua–Pickrell diffusions and Feller processes on the boundary of the graph of spectrahttps://projecteuclid.org/euclid.aihp/1584345637<strong>Theodoros Assiotis</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1251--1283.</p><p><strong>Abstract:</strong><br/>
We consider consistent diffusion dynamics, leaving the celebrated Hua–Pickrell measures, depending on a complex parameter $s$, invariant. These, give rise to Feller–Markov processes on the infinite dimensional boundary $\Omega $ of the “graph of spectra”, the continuum analogue of the Gelfand–Tsetlin graph, via the method of intertwiners of Borodin and Olshanski. In the particular case of $s=0$, this stochastic process is closely related to the $\mathsf{Sine_{2}}$ point process on $\mathbb{R}$ that describes the spectrum in the bulk of large random matrices. Equivalently, these coherent dynamics are associated to interlacing diffusions in Gelfand–Tsetlin patterns having certain Gibbs invariant measures. Moreover, under an application of the Cayley transform when $s=0$ we obtain processes on the circle leaving invariant the multilevel Circular Unitary Ensemble. We finally prove that the Feller processes on $\Omega $ corresponding to Dyson’s Brownian motion and its stationary analogue are given by explicit and very simple deterministic dynamical systems.
</p>projecteuclid.org/euclid.aihp/1584345637_20200316040027Mon, 16 Mar 2020 04:00 EDTOutliers in the spectrum for products of independent random matriceshttps://projecteuclid.org/euclid.aihp/1584345638<strong>Natalie Coston</strong>, <strong>Sean O’Rourke</strong>, <strong>Philip Matchett Wood</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1284--1320.</p><p><strong>Abstract:</strong><br/>
For fixed $m\geq 1$, we consider the product of $m$ independent $n\times n$ random matrices with iid entries as $n\to \infty $. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the $m$th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao ( Probab. Theory Related Fields 155 (2013) 231–263) for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.
</p>projecteuclid.org/euclid.aihp/1584345638_20200316040027Mon, 16 Mar 2020 04:00 EDTInteracting self-avoiding polygonshttps://projecteuclid.org/euclid.aihp/1584345639<strong>Volker Betz</strong>, <strong>Helge Schäfer</strong>, <strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1321--1335.</p><p><strong>Abstract:</strong><br/>
We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.
</p>projecteuclid.org/euclid.aihp/1584345639_20200316040027Mon, 16 Mar 2020 04:00 EDTLower bounds for fluctuations in first-passage percolation for general distributionshttps://projecteuclid.org/euclid.aihp/1584345640<strong>Michael Damron</strong>, <strong>Jack Hanson</strong>, <strong>Christian Houdré</strong>, <strong>Chen Xu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1336--1357.</p><p><strong>Abstract:</strong><br/>
In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice $\mathbb{Z}^{d}$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $\log \|x\|$. This result was found in the ’90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\log \|x\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of “hi-mode” (large).
</p>projecteuclid.org/euclid.aihp/1584345640_20200316040027Mon, 16 Mar 2020 04:00 EDTThe sharp phase transition for level set percolation of smooth planar Gaussian fieldshttps://projecteuclid.org/euclid.aihp/1584345641<strong>Stephen Muirhead</strong>, <strong>Hugo Vanneuville</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1358--1390.</p><p><strong>Abstract:</strong><br/>
We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two – roughly equivalent to the integrability of the covariance kernel – whereas previously the phase transition was only known in the case of the Bargmann–Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp , demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially.
Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.
</p>projecteuclid.org/euclid.aihp/1584345641_20200316040027Mon, 16 Mar 2020 04:00 EDTParameter recovery in two-component contamination mixtures: The $L^{2}$ strategyhttps://projecteuclid.org/euclid.aihp/1584345642<strong>Sébastien Gadat</strong>, <strong>Jonas Kahn</strong>, <strong>Clément Marteau</strong>, <strong>Cathy Maugis-Rabusseau</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1391--1418.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^{\star }=(1-\lambda^{\star })\phi +\lambda^{\star }\phi (\cdot-\mu^{\star })$, where the shape $\phi $ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(\lambda^{\star },\mu^{\star })\in (0,1)\times \mathbb{R}^{d}$. In particular, we prove that the classical parametric rate $1/\sqrt{n}$ cannot be reached when at least one of these parameters is allowed to tend to $0$ with $n$.
</p>projecteuclid.org/euclid.aihp/1584345642_20200316040027Mon, 16 Mar 2020 04:00 EDTSize of a minimal cutset in supercritical first passage percolationhttps://projecteuclid.org/euclid.aihp/1584345643<strong>Barbara Dembin</strong>, <strong>Marie Théret</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1419--1439.</p><p><strong>Abstract:</strong><br/>
We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z}^{d}$ given a distribution $G$ on $[0,+\infty ]$ (including $+\infty $). We suppose that $G(\{0\})>1-p_{c}(d)$, i.e. , the edges of positive passage time are in the subcritical regime of percolation on $\mathbb{Z}^{d}$. We consider a cylinder of basis an hyperrectangle of dimension $d-1$ whose sides have length $n$ and of height $h(n)$ with $h(n)$ negligible compared to $n$ ( i.e. , $h(n)/n\rightarrow 0$ when $n$ goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by $n^{d-1}$ converges towards the flow constant which is null in the case $G(\{0\})>1-p_{c}(d)$. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by $\psi_{n}$ the minimal cardinality of such a set of edges, we prove here that $\psi_{n}/n^{d-1}$ converges almost surely towards a constant.
</p>projecteuclid.org/euclid.aihp/1584345643_20200316040027Mon, 16 Mar 2020 04:00 EDTLocal densities for a class of degenerate diffusionshttps://projecteuclid.org/euclid.aihp/1584345644<strong>Alberto Lanconelli</strong>, <strong>Stefano Pagliarani</strong>, <strong>Andrea Pascucci</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1440--1464.</p><p><strong>Abstract:</strong><br/>
We study a class of $\mathbb{R}^{d}$-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Itô formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hörmander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.
</p>projecteuclid.org/euclid.aihp/1584345644_20200316040027Mon, 16 Mar 2020 04:00 EDTExponential weights in multivariate regression and a low-rankness favoring priorhttps://projecteuclid.org/euclid.aihp/1584345645<strong>Arnak S. Dalalyan</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1465--1483.</p><p><strong>Abstract:</strong><br/>
We establish theoretical guarantees for the expected prediction error of the exponentially weighted aggregate in the case of multivariate regression that is when the label vector is multidimensional. We consider the regression model with fixed design and bounded noise. The first new feature uncovered by our guarantees is that it is not necessary to require independence of the observations: a symmetry condition on the noise distribution alone suffices to get a sharp risk bound. This result needs the regression vectors to be bounded. A second curious finding concerns the case of unbounded regression vectors but independent noise. It turns out that applying exponential weights to the label vectors perturbed by a uniform noise leads to an estimator satisfying a sharp oracle inequality. The last contribution is the instantiation of the proposed oracle inequalities to problems in which the unknown parameter is a matrix. We propose a low-rankness favoring prior and show that it leads to an estimator that is optimal under weak assumptions.
</p>projecteuclid.org/euclid.aihp/1584345645_20200316040027Mon, 16 Mar 2020 04:00 EDTStein’s method for functions of multivariate normal random variableshttps://projecteuclid.org/euclid.aihp/1584345646<strong>Robert E. Gaunt</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56, Number 2, 1484--1513.</p><p><strong>Abstract:</strong><br/>
By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_{n})_{n\geq 1}$ converges in distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random variables $(g(\mathbf{W}_{n}))_{n\geq 1}$ converges in distribution to $g(\Sigma^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^{d}\rightarrow \mathbb{R}$ is continuous. In this paper, we develop Stein’s method for the problem of deriving explicit bounds on the distance between $g(\mathbf{W}_{n})$ and $g(\Sigma^{1/2} \mathbf{Z})$ with respect to smooth probability metrics. We obtain several bounds for the case that the $j$-component of $\mathbf{W}_{n}$ is given by $W_{n,j}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{ij}$, where the $X_{ij}$ are independent. In particular, provided $g$ satisfies certain differentiability and growth rate conditions, we obtain an order $n^{-(p-1)/2}$ bound, for smooth test functions, if the first $p$ moments of the $X_{ij}$ agree with those of the normal distribution. If $p$ is an even integer and $g$ is an even function, this convergence rate can be improved further to order $n^{-p/2}$. These convergence rates are shown to be of optimal order. We apply our general bounds to some examples, which include the distributional approximation of asymptotically chi-square distributed statistics; the approximation of expectations of smooth functions of binomial and Poisson random variables; rates of convergence in the delta method; and a quantitative variance-gamma approximation of the $D_{2}^{*}$ statistic for alignment-free sequence comparison in the case of binary sequences.
</p>projecteuclid.org/euclid.aihp/1584345646_20200316040027Mon, 16 Mar 2020 04:00 EDT