Annales de l'Institut Henri Poincaré, Probabilités et Statistiques Articles (Project Euclid)
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The latest articles from Annales de l'Institut Henri Poincaré, Probabilités et Statistiques on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 23 Mar 2011 09:35 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject 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 EDTFluctuations of bridges, reciprocal characteristics and concentration of measurehttps://projecteuclid.org/euclid.aihp/1531296025<strong>Giovanni Conforti</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1432--1463.</p><p><strong>Abstract:</strong><br/>
Conditions on the generator of a Markov process to control the fluctuations of its bridges are found. In particular, continuous time random walks on graphs and gradient diffusions are considered. Under these conditions, a concentration of measure inequality for the marginals of the bridge of a gradient diffusion and refined large deviation expansions for the tails of a random walk on a graph are derived. In contrast with the existing literature about bridges, all the estimates we obtain hold for non asymptotic time scales. New concentration of measure inequalities for pinned Poisson random vectors are also established. The quantities expressing our conditions are the so called reciprocal characteristics associated with the Markov generator.
</p>projecteuclid.org/euclid.aihp/1531296025_20180711040025Wed, 11 Jul 2018 04:00 EDTConstruction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada–Watanabe principlehttps://projecteuclid.org/euclid.aihp/1531296026<strong>David R. Baños</strong>, <strong>Sindre Duedahl</strong>, <strong>Thilo Meyer-Brandis</strong>, <strong>Frank Proske</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1464--1491.</p><p><strong>Abstract:</strong><br/>
In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.
</p>projecteuclid.org/euclid.aihp/1531296026_20180711040025Wed, 11 Jul 2018 04:00 EDTThick points of high-dimensional Gaussian free fieldshttps://projecteuclid.org/euclid.aihp/1531296027<strong>Linan Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1492--1526.</p><p><strong>Abstract:</strong><br/>
This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions. Under this setting, we introduce the definition of thick points which, heuristically speaking, are points where the value of the Gaussian Free Field is unusually large. We then establish a result on the Hausdorff dimension of the sets containing thick points.
</p>projecteuclid.org/euclid.aihp/1531296027_20180711040025Wed, 11 Jul 2018 04:00 EDTThe size of the last merger and time reversal in $\Lambda$-coalescentshttps://projecteuclid.org/euclid.aihp/1531296028<strong>Götz Kersting</strong>, <strong>Jason Schweinsberg</strong>, <strong>Anton Wakolbinger</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1527--1555.</p><p><strong>Abstract:</strong><br/>
We consider the number of blocks involved in the last merger of a $\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n\to\infty$, the sequence of these random variables (a) is tight, (b) converges in distribution to a finite random variable or (c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of invariant measures for the dynamics of the block-counting process, and in case (b) investigate the time-reversal of the block-counting process back from the time of the last merger.
</p>projecteuclid.org/euclid.aihp/1531296028_20180711040025Wed, 11 Jul 2018 04:00 EDTOptimal discretization of stochastic integrals driven by general Brownian semimartingalehttps://projecteuclid.org/euclid.aihp/1531296029<strong>Emmanuel Gobet</strong>, <strong>Uladzislau Stazhynski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1556--1582.</p><p><strong>Abstract:</strong><br/>
We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.
</p>projecteuclid.org/euclid.aihp/1531296029_20180711040025Wed, 11 Jul 2018 04:00 EDTLow-rank diffusion matrix estimation for high-dimensional time-changed Lévy processeshttps://projecteuclid.org/euclid.aihp/1531296030<strong>Denis Belomestny</strong>, <strong>Mathias Trabs</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1583--1621.</p><p><strong>Abstract:</strong><br/>
The estimation of the diffusion matrix $\Sigma$ of a high-dimensional, possibly time-changed Lévy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $\Sigma$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $\Sigma$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.
</p>projecteuclid.org/euclid.aihp/1531296030_20180711040025Wed, 11 Jul 2018 04:00 EDTThe near-critical Gibbs measure of the branching random walkhttps://projecteuclid.org/euclid.aihp/1531296031<strong>Michel Pain</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1622--1666.</p><p><strong>Abstract:</strong><br/>
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $\nu_{n,\beta}$ on the $n$th generation, which is also the polymer measure on a disordered tree with inverse temperature $\beta$. The convergence of the partition function $W_{n,\beta}$, after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi ( Ann. Probab. 42 (3) (2014) 959–993) in the critical case $\beta=1$ and by Madaule ( J. Theoret. Probab. 30 (1) (2017) 27–63) when $\beta>1$. We study here the near-critical case, where $\beta_{n}\to1$, and prove the convergence of $W_{n,\beta_{n}}$, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure $\nu_{n,\beta}$ have been studied by Madaule ( Stochastic Process. Appl. 126 (2) (2016) 470–502) in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.
</p>projecteuclid.org/euclid.aihp/1531296031_20180711040025Wed, 11 Jul 2018 04:00 EDTCharacterization of a class of weak transport-entropy inequalities on the linehttps://projecteuclid.org/euclid.aihp/1531296032<strong>Nathael Gozlan</strong>, <strong>Cyril Roberto</strong>, <strong>Paul-Marie Samson</strong>, <strong>Yan Shu</strong>, <strong>Prasad Tetali</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1667--1693.</p><p><strong>Abstract:</strong><br/>
We study an weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.
</p>projecteuclid.org/euclid.aihp/1531296032_20180711040025Wed, 11 Jul 2018 04:00 EDTLiouville quantum gravity on the unit diskhttps://projecteuclid.org/euclid.aihp/1531296033<strong>Yichao Huang</strong>, <strong>Rémi Rhodes</strong>, <strong>Vincent Vargas</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1694--1730.</p><p><strong>Abstract:</strong><br/>
Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk.
</p>projecteuclid.org/euclid.aihp/1531296033_20180711040025Wed, 11 Jul 2018 04:00 EDTInterpolation process between standard diffusion and fractional diffusionhttps://projecteuclid.org/euclid.aihp/1531296034<strong>Cédric Bernardin</strong>, <strong>Patrícia Gonçalves</strong>, <strong>Milton Jara</strong>, <strong>Marielle Simon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 3, 1731--1757.</p><p><strong>Abstract:</strong><br/>
We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume ( Nonlinearity 25 (4) (2012) 1099–1133; Arch. Ration. Mech. Anal. 220 (2) (2016) 505–542). We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein–Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric $3/2$-stable Lévy process. This result extends and solves a problem left open in ( J. Stat. Phys. 159 (6) (2015) 1327–1368).
</p>projecteuclid.org/euclid.aihp/1531296034_20180711040025Wed, 11 Jul 2018 04:00 EDTGaussian fluctuations for the classical XY modelhttps://projecteuclid.org/euclid.aihp/1539849782<strong>Charles M. Newman</strong>, <strong>Wei Wu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1759--1777.</p><p><strong>Abstract:</strong><br/>
We study the classical XY model in bounded domains of $\mathbb{Z}^{d}$ with Dirichlet boundary conditions. We prove that when the temperature goes to zero faster than a certain rate as the lattice spacing goes to zero, the fluctuation field converges to a Gaussian white noise. This and related results also apply to a large class of gradient field models.
</p>projecteuclid.org/euclid.aihp/1539849782_20181018040325Thu, 18 Oct 2018 04:03 EDTContinuum percolation in high dimensionshttps://projecteuclid.org/euclid.aihp/1539849783<strong>Jean-Baptiste Gouéré</strong>, <strong>Régine Marchand</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1778--1804.</p><p><strong>Abstract:</strong><br/>
Consider a Boolean model $\Sigma$ in $\mathbb{R}^{d}$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d. with common distribution $\nu$. The critical covered volume is the proportion of space covered by $\Sigma$ when the intensity $\lambda$ is critical for percolation. We study the asymptotic behaviour, as $d$ tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension.
</p>projecteuclid.org/euclid.aihp/1539849783_20181018040325Thu, 18 Oct 2018 04:03 EDTConvergence to equilibrium in the free Fokker–Planck equation with a double-well potentialhttps://projecteuclid.org/euclid.aihp/1539849784<strong>Catherine Donati-Martin</strong>, <strong>Benjamin Groux</strong>, <strong>Mylène Maïda</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1805--1818.</p><p><strong>Abstract:</strong><br/>
We consider the one-dimensional free Fokker–Planck equation
\[\frac{\partial \mu_{t}}{\partial t}=\frac{\partial }{\partial x}[\mu_{t}\cdot (\frac{1}{2}V'-H\mu_{t})],\] where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x)=\frac{1}{4}x^{4}+\frac{c}{2}x^{2}$, with $c\ge -2$. We prove that the solution $(\mu_{t})_{t\ge 0}$ of this PDE converges in Wasserstein distance of any order $p\ge 1$ to the equilibrium measure $\mu_{V}$ as $t$ goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques.
</p>projecteuclid.org/euclid.aihp/1539849784_20181018040325Thu, 18 Oct 2018 04:03 EDTPercolation and isoperimetry on roughly transitive graphshttps://projecteuclid.org/euclid.aihp/1539849785<strong>Elisabetta Candellero</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1819--1847.</p><p><strong>Abstract:</strong><br/>
In this paper we study percolation on a roughly transitive graph $G$ with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that $p_{c}<1$, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When $G$ is transitive, the fact that $p_{c}<1$ was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov’s theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.
</p>projecteuclid.org/euclid.aihp/1539849785_20181018040325Thu, 18 Oct 2018 04:03 EDTMulti-arm incipient infinite clusters in 2D: Scaling limits and winding numbershttps://projecteuclid.org/euclid.aihp/1539849786<strong>Chang-Long Yao</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1848--1876.</p><p><strong>Abstract:</strong><br/>
We study the alternating $k$-arm incipient infinite cluster (IIC) of site percolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman’s result that the scaling limit of critical site percolation on $\mathbb{T}$ is CLE$_{6}$, we prove the existence of the scaling limit of the $k$-arm IIC for $k=1,2,4$. Conditioned on the event that there are open and closed arms connecting the origin to $\partial\mathbb{D}_{R}$, we show that the winding number variance of the arms is $(3/2+o(1))\log R$ as $R\rightarrow\infty$, which confirms a prediction of Wieland and Wilson [ Phys. Rev. E 68 (2003) 056101]. Our proof uses two-sided radial SLE$_{6}$ and coupling argument. Using this result we get an explicit form for the CLT of the winding numbers, and get analogous result for the 2-arm IIC, thus improving our earlier result.
</p>projecteuclid.org/euclid.aihp/1539849786_20181018040325Thu, 18 Oct 2018 04:03 EDTBrownian motion and random walk above quenched random wallhttps://projecteuclid.org/euclid.aihp/1539849787<strong>Bastien Mallein</strong>, <strong>Piotr Miłoś</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1877--1916.</p><p><strong>Abstract:</strong><br/>
We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_{n}\}$ and $\{W_{n}\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N}B_{n}\geq W_{n}\vert W)=N^{-\gamma+o(1)}$ for a non-random $\gamma\geq1/2$. In the classical setting, $W_{n}\equiv0$, it is well-known that $\gamma=1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\operatorname{Var}(B_{1})/\operatorname{Var}(W_{1})$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein–Uhlenbeck processes. In the latter case the probability decays at exponential rate.
</p>projecteuclid.org/euclid.aihp/1539849787_20181018040325Thu, 18 Oct 2018 04:03 EDTMesoscopic central limit theorem for general $\beta$-ensembleshttps://projecteuclid.org/euclid.aihp/1539849788<strong>Florent Bekerman</strong>, <strong>Asad Lodhia</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1917--1938.</p><p><strong>Abstract:</strong><br/>
We prove that the linear statistics of eigenvalues of $\beta$-log gases satisfying the one-cut and off-critical assumption with a potential $V\in C^{7}(\mathbb{R})$ satisfy a central limit theorem at all mesoscopic scales $\alpha\in(0;1)$. We prove this for compactly supported test functions $f\in C^{6}(\mathbb{R})$ using loop equations at all orders along with rigidity estimates.
</p>projecteuclid.org/euclid.aihp/1539849788_20181018040325Thu, 18 Oct 2018 04:03 EDTScaling limits of stochastic processes associated with resistance formshttps://projecteuclid.org/euclid.aihp/1539849789<strong>D. A. Croydon</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1939--1968.</p><p><strong>Abstract:</strong><br/>
We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov–Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. We further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.
</p>projecteuclid.org/euclid.aihp/1539849789_20181018040325Thu, 18 Oct 2018 04:03 EDTGlobal well-posedness of complex Ginzburg–Landau equation with a space–time white noisehttps://projecteuclid.org/euclid.aihp/1539849790<strong>Masato Hoshino</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 1969--2001.</p><p><strong>Abstract:</strong><br/>
We show the global-in-time well-posedness of the complex Ginzburg–Landau (CGL) equation with a space–time white noise on the 3-dimensional torus. Our method is based on Mourrat and Weber (Global well-posedness of the dynamic $\Phi_{3}^{4}$ model on the torus), where Mourrat and Weber showed the global well-posedness for the dynamical $\Phi_{3}^{4}$ model. We prove a priori $L^{2p}$ estimate for the paracontrolled solution as in the deterministic case [ Phys. D 71 (1994) 285–318].
</p>projecteuclid.org/euclid.aihp/1539849790_20181018040325Thu, 18 Oct 2018 04:03 EDTLocal large deviations principle for occupation measures of the stochastic damped nonlinear wave equationhttps://projecteuclid.org/euclid.aihp/1539849791<strong>D. Martirosyan</strong>, <strong>V. Nersesyan</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2002--2041.</p><p><strong>Abstract:</strong><br/>
We consider the damped nonlinear wave (NLW) equation driven by a noise which is white in time and colored in space. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and is a novelty in the context of randomly forced PDE’s. The proof is based on an extension of methods developed in ( Comm. Pure Appl. Math. 68 (12) (2015) 2108–2143) and (Large deviations and mixing for dissipative PDE’s with unbounded random kicks (2014) Preprint) in the case of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system and the complex Ginzburg–Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere).
</p>projecteuclid.org/euclid.aihp/1539849791_20181018040325Thu, 18 Oct 2018 04:03 EDTMultifractality of jump diffusion processeshttps://projecteuclid.org/euclid.aihp/1539849792<strong>Xiaochuan Yang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2042--2074.</p><p><strong>Abstract:</strong><br/>
We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral et al. who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass’ stable-like processes and non-degenerate stable-driven SDEs.
</p>projecteuclid.org/euclid.aihp/1539849792_20181018040325Thu, 18 Oct 2018 04:03 EDTA characterization of a class of convex log-Sobolev inequalities on the real linehttps://projecteuclid.org/euclid.aihp/1539849793<strong>Yan Shu</strong>, <strong>Michał Strzelecki</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2075--2091.</p><p><strong>Abstract:</strong><br/>
We give a sufficient and necessary condition for a probability measure $\mu$ on the real line to satisfy the logarithmic Sobolev inequality for convex functions. The condition is expressed in terms of the unique left-continuous and non-decreasing map transporting the symmetric exponential measure onto $\mu$. The main tool in the proof is the theory of weak transport costs.
As a consequence, we obtain dimension-free concentration bounds for the lower and upper tails of convex functions of independent random variables which satisfy the convex log-Sobolev inequality.
</p>projecteuclid.org/euclid.aihp/1539849793_20181018040325Thu, 18 Oct 2018 04:03 EDTIsoperimetry in supercritical bond percolation in dimensions three and higherhttps://projecteuclid.org/euclid.aihp/1539849794<strong>Julian Gold</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2092--2158.</p><p><strong>Abstract:</strong><br/>
We study the isoperimetric subgraphs of the infinite cluster $\mathbf{C}_{\infty}$ for supercritical bond percolation on $\mathbb{Z}^{d}$ with $d\geq3$. Specifically, we consider subgraphs of $\mathbf{C}_{\infty}\cap[-n,n]^{d}$ having minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs: when suitably rescaled, they converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\mathbf{C}_{\infty}\cap[-n,n]^{d}$, settling a conjecture of Benjamini for the version of the Cheeger constant defined here.
</p>projecteuclid.org/euclid.aihp/1539849794_20181018040325Thu, 18 Oct 2018 04:03 EDTClassical and quantum part of the environment for quantum Langevin equationshttps://projecteuclid.org/euclid.aihp/1539849795<strong>Stéphane Attal</strong>, <strong>Ivan Bardet</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2159--2176.</p><p><strong>Abstract:</strong><br/>
Among quantum Langevin equations describing the unitary time evolution of a quantum system in contact with a quantum bath, we completely characterize those equations which are actually driven by classical noises. The characterization is purely algebraic, in terms of the coefficients of the equation. In a second part, we consider general quantum Langevin equations and we prove that they can always be split into a maximal part driven by classical noises and a purely quantum one.
</p>projecteuclid.org/euclid.aihp/1539849795_20181018040325Thu, 18 Oct 2018 04:03 EDTHow can a clairvoyant particle escape the exclusion process?https://projecteuclid.org/euclid.aihp/1539849796<strong>Rangel Baldasso</strong>, <strong>Augusto Teixeira</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2177--2202.</p><p><strong>Abstract:</strong><br/>
We study a detection problem in the following setting: On the one-dimensional integer lattice, at time zero, place detectors on each site independently with probability $\rho \in{[0,1)}$ and let they evolve as a simple symmetric exclusion process. At time zero, place a target at the origin. The target moves only at integer times, and can move to any site that is within distance $R$ from its current position. Assume also that the target can predict the future movement of all detectors. We prove that, for $R$ large enough (depending on the value of $\rho $) it is possible for the target to avoid detection forever with positive probability. The proof of this result uses two ingredients of independent interest. First we establish a renormalisation scheme that can be used to prove percolation for dependent oriented models under a certain decoupling condition. This result is general and does not rely on the specifities of the model. As an application, we prove our main theorem for different dynamics, such as independent random walks and independent renewal chains. We also prove existence of oriented percolation for random interlacements and for its vacant set for large dimensions. The second step of the proof is a space–time decoupling for the exclusion process.
</p>projecteuclid.org/euclid.aihp/1539849796_20181018040325Thu, 18 Oct 2018 04:03 EDTThe geometry of a critical percolation cluster on the UIPThttps://projecteuclid.org/euclid.aihp/1539849797<strong>Matthias Gorny</strong>, <strong>Édouard Maurel-Segala</strong>, <strong>Arvind Singh</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2203--2238.</p><p><strong>Abstract:</strong><br/>
We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differ by a factor $2$ from those computed previously by Angel and Curien [ Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 405–431] in the case of critical site percolation on the uniform infinite half-plane triangulation.
</p>projecteuclid.org/euclid.aihp/1539849797_20181018040325Thu, 18 Oct 2018 04:03 EDTOn the large deviations of traces of random matriceshttps://projecteuclid.org/euclid.aihp/1539849798<strong>Fanny Augeri</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2239--2285.</p><p><strong>Abstract:</strong><br/>
We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta $-ensembles in three cases: the case of $\beta $-ensembles associated with a convex potential with polynomial growth, the case of Gaussian Wigner matrices, and the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^{\alpha }}$, for some constant $c>0$ and with $\alpha \in (0,2)$.
</p>projecteuclid.org/euclid.aihp/1539849798_20181018040325Thu, 18 Oct 2018 04:03 EDTTransporting random measures on the line and embedding excursions into Brownian motionhttps://projecteuclid.org/euclid.aihp/1539849799<strong>Günter Last</strong>, <strong>Wenpin Tang</strong>, <strong>Hermann Thorisson</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2286--2303.</p><p><strong>Abstract:</strong><br/>
We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on the real line $\mathbb{R}$ with equal intensities. An allocation is an equivariant random mapping from $\mathbb{R}$ to $\mathbb{R}$. We give sufficient and partially necessary conditions for the existence of allocations transporting $\xi$ to $\eta$. An important ingredient of our approach is a transport kernel balancing $\xi$ and $\eta$, provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on $(-\infty,0]$, an excursion distributed according to a conditional Itô measure and a Brownian motion starting after this excursion. An analogous result holds for Bismut’s excursion measure.
</p>projecteuclid.org/euclid.aihp/1539849799_20181018040325Thu, 18 Oct 2018 04:03 EDTA temporal central limit theorem for real-valued cocycles over rotationshttps://projecteuclid.org/euclid.aihp/1539849800<strong>Michael Bromberg</strong>, <strong>Corinna Ulcigrai</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2304--2334.</p><p><strong>Abstract:</strong><br/>
We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $\alpha$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $\beta$. When $\alpha$ is badly approximable and $\beta$ is badly approximable with respect to $\alpha$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D. Dolgopyat and O. Sarig), namely we show that for any fixed initial point, the occupancy random variables , suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $\alpha$ is quadratic irrational, $\beta$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila–Dolgopyat–Duryev–Sarig ( Israel J. Math. 207 (2015) 653–717) and Dolgopyat–Sarig ( J. Stat. Phys. 166 (2017) 680–713). We also use renormalization, but in order to treat irrational values of $\beta$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.
</p>projecteuclid.org/euclid.aihp/1539849800_20181018040325Thu, 18 Oct 2018 04:03 EDTKinetically constrained lattice gases: Tagged particle diffusionhttps://projecteuclid.org/euclid.aihp/1539849801<strong>O. Blondel</strong>, <strong>C. Toninelli</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2335--2348.</p><p><strong>Abstract:</strong><br/>
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb{Z}^{d}$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for all dimensions $d\geq2$ and for any equilibrium particle density, under diffusive rescaling the motion of the tracer converges to a $d$-dimensional Brownian motion with non-degenerate diffusion matrix. Therefore we disprove the occurrence of a diffusive/non diffusive transition which had been conjectured in physics literature. Our technique is flexible enough and can be extended to analyse the tracer behavior for other choices of constraints.
</p>projecteuclid.org/euclid.aihp/1539849801_20181018040325Thu, 18 Oct 2018 04:03 EDTLocation of the path supremum for self-similar processes with stationary incrementshttps://projecteuclid.org/euclid.aihp/1539849802<strong>Yi Shen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54, Number 4, 2349--2360.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. A point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. An upper bound for the value of the density function is established. We further discuss self-similar Lévy processes as an example. Most of the results in this paper can be generalized to a group of random locations, including the location of the largest jump, etc.
</p>projecteuclid.org/euclid.aihp/1539849802_20181018040325Thu, 18 Oct 2018 04:03 EDTScaling limits for the critical Fortuin–Kasteleyn model on a random planar map I: Cone timeshttps://projecteuclid.org/euclid.aihp/1547802394<strong>Ewain Gwynne</strong>, <strong>Cheng Mao</strong>, <strong>Xin Sun</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 1--60.</p><p><strong>Abstract:</strong><br/>
Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin–Kasteleyn (FK) model. He showed that a certain two-dimensional random walk associated with the infinite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or “flexible orders”) in the inventory accumulation model converge in the scaling limit to the $\pi/2$-cone times of the correlated Brownian motion. This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE loops on an independent Liouville quantum gravity surface.
</p>projecteuclid.org/euclid.aihp/1547802394_20190118040648Fri, 18 Jan 2019 04:06 ESTOn the fourth moment condition for Rademacher chaoshttps://projecteuclid.org/euclid.aihp/1547802395<strong>Christian Döbler</strong>, <strong>Kai Krokowski</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 61--97.</p><p><strong>Abstract:</strong><br/>
Adapting the spectral viewpoint suggested in ( Ann. Probab. 40 (6) (2012) 2439–2459) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure ( Ann. Probab. (2017)), we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e. non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order $m\geq2$ with respect to a symmetric Rademacher sequence. This behavior, which is in contrast to the Gaussian ( Ann. Probab. 33 (1) (2005) 177–193) and Poisson ( Ann. Probab. (2017)) situation, closely resembles that of degenerate, non-symmetric $U$-statistics from the classical paper ( J. Multivariate Anal. 34 (2) (1990) 275–289).
</p>projecteuclid.org/euclid.aihp/1547802395_20190118040648Fri, 18 Jan 2019 04:06 ESTProducts of random matrices from polynomial ensembleshttps://projecteuclid.org/euclid.aihp/1547802396<strong>Mario Kieburg</strong>, <strong>Holger Kösters</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 98--126.</p><p><strong>Abstract:</strong><br/>
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and generalize a number of recent results in random matrix theory, including a transformation formula for the kernels of the corresponding determinantal point processes. Starting from these results, we construct a continuous family of random matrix ensembles interpolating between the products of different numbers of Ginibre matrices and inverse Ginibre matrices. Furthermore, we make contact to the asymptotic distribution of the Lyapunov exponents of the products of a large number of bi-unitarily invariant random matrices of fixed dimension.
</p>projecteuclid.org/euclid.aihp/1547802396_20190118040648Fri, 18 Jan 2019 04:06 ESTBarrier estimates for a critical Galton–Watson process and the cover time of the binary treehttps://projecteuclid.org/euclid.aihp/1547802397<strong>David Belius</strong>, <strong>Jay Rosen</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 127--154.</p><p><strong>Abstract:</strong><br/>
For the critical Galton–Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two-dimensional manifolds. As an application of the barrier estimates, we prove that if $C_{L}$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_{L}/2^{L+1}}-\sqrt{2\log2}L+\log L/\sqrt{2\log2}$ is tight. The latter improves results of Aldous ( J. Math. Anal. Appl. 157 (1991) 271–283), Bramson and Zeitouni ( Ann. Probab. 37 (2009) 615–653) and Ding and Zeitouni ( Stochastic Process. Appl. 122 (2012) 2117–2133). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for compact two-dimensional manifolds.
</p>projecteuclid.org/euclid.aihp/1547802397_20190118040648Fri, 18 Jan 2019 04:06 ESTLocal limits of large Galton–Watson trees rerooted at a random vertexhttps://projecteuclid.org/euclid.aihp/1547802398<strong>Benedikt Stufler</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 155--183.</p><p><strong>Abstract:</strong><br/>
We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton–Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.
</p>projecteuclid.org/euclid.aihp/1547802398_20190118040648Fri, 18 Jan 2019 04:06 ESTBranching diffusion representation of semilinear PDEs and Monte Carlo approximationhttps://projecteuclid.org/euclid.aihp/1547802399<strong>Pierre Henry-Labordère</strong>, <strong>Nadia Oudjane</strong>, <strong>Xiaolu Tan</strong>, <strong>Nizar Touzi</strong>, <strong>Xavier Warin</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 184--210.</p><p><strong>Abstract:</strong><br/>
We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [ Theory Probab. Appl. 9 (1964) 445–449], Watanabe [ J. Math. Kyoto Univ. 4 (1965) 385–398] and McKean [ Comm. Pure Appl. Math. 28 (1975) 323–331], by allowing for polynomial nonlinearity in the pair $(u,Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to “small maturity” or “small nonlinearity” of the PDE. Our main ingredient is the Malliavin automatic differentiation technique as in [ Ann. Appl. Probab. 27 (2017) 3305–3341], based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.
</p>projecteuclid.org/euclid.aihp/1547802399_20190118040648Fri, 18 Jan 2019 04:06 ESTLarge deviations for the two-dimensional stochastic Navier–Stokes equation with vanishing noise correlationhttps://projecteuclid.org/euclid.aihp/1547802400<strong>Sandra Cerrai</strong>, <strong>Arnaud Debussche</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 211--236.</p><p><strong>Abstract:</strong><br/>
We are dealing with the validity of a large deviation principle for the two-dimensional Navier–Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\epsilon$ and $\delta(\epsilon)$, respectively, with $0<\epsilon,\delta(\epsilon)\ll1$. Depending on the relationship between $\epsilon$ and $\delta(\epsilon)$ we will prove the validity of the large deviation principle in different functional spaces.
</p>projecteuclid.org/euclid.aihp/1547802400_20190118040648Fri, 18 Jan 2019 04:06 ESTBrownian disks and the Brownian snakehttps://projecteuclid.org/euclid.aihp/1547802401<strong>Jean-François Le Gall</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 237--313.</p><p><strong>Abstract:</strong><br/>
We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a continuous random tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications of our construction. In particular, we prove that the uniform measure on the boundary can be obtained as the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. We also prove that connected components of the complement of the Brownian net are Brownian disks, as it was suggested in the recent work of Miller and Sheffield. Finally, we show that connected components of the complement of balls centered at the distinguished point of the Brownian map are independent Brownian disks, conditionally on their volumes and perimeters.
</p>projecteuclid.org/euclid.aihp/1547802401_20190118040648Fri, 18 Jan 2019 04:06 ESTConditioning a Brownian loop-soup cluster on a portion of its boundaryhttps://projecteuclid.org/euclid.aihp/1547802402<strong>Wei Qian</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 314--340.</p><p><strong>Abstract:</strong><br/>
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $\partial $ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $\partial $ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $\partial $ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at $c=14/15$ for the connectedness of the loops that touch $\partial $.
Our results can be viewed as an extension of some of the results in our paper ( J. Eur. Math. Soc. (2019) to appear) in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained in ( J. Eur. Math. Soc. (2019) to appear) that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.
</p>projecteuclid.org/euclid.aihp/1547802402_20190118040648Fri, 18 Jan 2019 04:06 ESTIntertwinings and Stein’s magic factors for birth–death processeshttps://projecteuclid.org/euclid.aihp/1547802403<strong>Bertrand Cloez</strong>, <strong>Claire Delplancke</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 341--377.</p><p><strong>Abstract:</strong><br/>
This article investigates second order intertwinings between semigroups of birth–death processes and discrete gradients on $\mathbb{N}$. It goes one step beyond a recent work of Chafaï and Joulin which establishes and applies to the analysis of birth–death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth–death and Feynman–Kac semigroups and weighted gradients on $\mathbb{N}$, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of discrete distributions. To illustrate the relevance of this approach, we also derive approximation results for the mixture of Poisson and geometric laws.
</p>projecteuclid.org/euclid.aihp/1547802403_20190118040648Fri, 18 Jan 2019 04:06 ESTMixing and decorrelation in infinite measure: The case of the periodic Sinai billiardhttps://projecteuclid.org/euclid.aihp/1547802404<strong>Françoise Pène</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 378--411.</p><p><strong>Abstract:</strong><br/>
We investigate the question of the rate of mixing for observables of a $\mathbb{Z}^{d}$-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main motivation of this article is the study of mixing rates for smooth observables of the $\mathbb{Z}^{2}$-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, enabling the study of the mixing rate even for observables with null integrals. This result is related to an Edgeworth expansion in the local limit theorem.
</p>projecteuclid.org/euclid.aihp/1547802404_20190118040648Fri, 18 Jan 2019 04:06 ESTThe local limit of random sorting networkshttps://projecteuclid.org/euclid.aihp/1547802405<strong>Omer Angel</strong>, <strong>Duncan Dauvergne</strong>, <strong>Alexander E. Holroyd</strong>, <strong>Bálint Virág</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 412--440.</p><p><strong>Abstract:</strong><br/>
A sorting network is a geodesic path from $12\cdots n$ to $n\cdots21$ in the Cayley graph of $S_{n}$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space–time locations of transpositions in a neighbourhood of $an$ for $a\in [0,1]$ as $n\to \infty $. Here time is scaled by a factor of $1/n$ and space is not scaled.
The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$.
To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.
</p>projecteuclid.org/euclid.aihp/1547802405_20190118040648Fri, 18 Jan 2019 04:06 ESTFinite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembleshttps://projecteuclid.org/euclid.aihp/1547802406<strong>Gernot Akemann</strong>, <strong>Tomasz Checinski</strong>, <strong>Dang-Zheng Liu</strong>, <strong>Eugene Strahov</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 441--479.</p><p><strong>Abstract:</strong><br/>
We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.
</p>projecteuclid.org/euclid.aihp/1547802406_20190118040648Fri, 18 Jan 2019 04:06 ESTFunctional limit theorem for the self-intersection local time of the fractional Brownian motionhttps://projecteuclid.org/euclid.aihp/1547802407<strong>Arturo Jaramillo</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 480--527.</p><p><strong>Abstract:</strong><br/>
Let $\{B_{t}\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as
\[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] where $p_{\varepsilon}(x)$ is the heat kernel. We prove that the process $\{I_{T}^{\varepsilon}-\mathbb{E}[I_{T}^{\varepsilon}]\}_{T\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\frac{3}{2d}<H\leq\frac{3}{4}$ and to a multiple of a sum of independent Hermite processes for $\frac{3}{4}<H<1$, in the space $C[0,\infty)$, endowed with the topology of uniform convergence on compacts.
</p>projecteuclid.org/euclid.aihp/1547802407_20190118040648Fri, 18 Jan 2019 04:06 ESTUniversality of Ghirlanda–Guerra identities and spin distributions in mixed $p$-spin modelshttps://projecteuclid.org/euclid.aihp/1547802408<strong>Yu-Ting Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 528--550.</p><p><strong>Abstract:</strong><br/>
We prove universality of the Ghirlanda–Guerra identities and spin distributions in the mixed $p$-spin models. The assumption for the universality of the identities requires exactly that the coupling constants have zero means and finite variances, and the result applies to the Sherrington–Kirkpatrick model. As an application, we obtain weakly convergent universality of spin distributions in the generic $p$-spin models under the condition of two matching moments. In particular, certain identities for 3-overlaps and 4-overlaps under the Gaussian disorder follow. Under the stronger mode of total variation convergence, we find that universality of spin distributions in the mixed $p$-spin models holds if mild dilution of connectivity by the Viana–Bray diluted spin glass Hamiltonians is present and the first three moments of coupling constants in the mixed $p$-spin Hamiltonians match. These universality results are in stark contrast to the characterization of spin distributions in the undiluted mixed $p$-spin models, which is known up to now that four matching moments are required in general.
</p>projecteuclid.org/euclid.aihp/1547802408_20190118040648Fri, 18 Jan 2019 04:06 ESTConvergence of the free Boltzmann quadrangulation with simple boundary to the Brownian diskhttps://projecteuclid.org/euclid.aihp/1547802409<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 551--589.</p><p><strong>Abstract:</strong><br/>
We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov–Hausdorff–Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov–Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.
</p>projecteuclid.org/euclid.aihp/1547802409_20190118040648Fri, 18 Jan 2019 04:06 ESTErgodicity of a system of interacting random walks with asymmetric interactionhttps://projecteuclid.org/euclid.aihp/1547802410<strong>Luisa Andreis</strong>, <strong>Amine Asselah</strong>, <strong>Paolo Dai Pra</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 1, 590--606.</p><p><strong>Abstract:</strong><br/>
We study $N$ interacting random walks on the positive integers. Each particle has drift $\delta$ towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.
</p>projecteuclid.org/euclid.aihp/1547802410_20190118040648Fri, 18 Jan 2019 04:06 ESTThe infinite Atlas process: Convergence to equilibriumhttps://projecteuclid.org/euclid.aihp/1557820825<strong>Amir Dembo</strong>, <strong>Milton Jara</strong>, <strong>Stefano Olla</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 607--619.</p><p><strong>Abstract:</strong><br/>
The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.
</p>projecteuclid.org/euclid.aihp/1557820825_20190514040050Tue, 14 May 2019 04:00 EDTExponential functionals of spectrally one-sided Lévy processes conditioned to stay positivehttps://projecteuclid.org/euclid.aihp/1557820826<strong>Grégoire Véchambre</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 620--660.</p><p><strong>Abstract:</strong><br/>
We study the properties of the exponential functional $\int_{0}^{+\infty }e^{-X^{\uparrow }(t)}\,dt$ where $X^{\uparrow }$ is a spectrally one-sided Lévy process conditioned to stay positive. In particular, we study finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at $0$ and smoothness of the density.
</p>projecteuclid.org/euclid.aihp/1557820826_20190514040050Tue, 14 May 2019 04:00 EDTLocation of the spectrum of Kronecker random matriceshttps://projecteuclid.org/euclid.aihp/1557820827<strong>Johannes Alt</strong>, <strong>László Erdős</strong>, <strong>Torben Krüger</strong>, <strong>Yuriy Nemish</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 661--696.</p><p><strong>Abstract:</strong><br/>
For a general class of large non-Hermitian random block matrices ${\boldsymbol X}$ we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of ${\boldsymbol X}$ as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from ( Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.
</p>projecteuclid.org/euclid.aihp/1557820827_20190514040050Tue, 14 May 2019 04:00 EDTHeat kernel estimates for anomalous heavy-tailed random walkshttps://projecteuclid.org/euclid.aihp/1557820828<strong>Mathav Murugan</strong>, <strong>Laurent Saloff-Coste</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 697--719.</p><p><strong>Abstract:</strong><br/>
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies’ perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.
</p>projecteuclid.org/euclid.aihp/1557820828_20190514040050Tue, 14 May 2019 04:00 EDTOn the exit time and stochastic homogenization of isotropic diffusions in large domainshttps://projecteuclid.org/euclid.aihp/1557820829<strong>Benjamin Fehrman</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 720--755.</p><p><strong>Abstract:</strong><br/>
Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni ( Invent. Math. 164 (2006) 455–567), upon whose results the present work relies strongly.
</p>projecteuclid.org/euclid.aihp/1557820829_20190514040050Tue, 14 May 2019 04:00 EDTOn the spectral gap of spherical spin glass dynamicshttps://projecteuclid.org/euclid.aihp/1557820830<strong>Reza Gheissari</strong>, <strong>Aukosh Jagannath</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 756--776.</p><p><strong>Abstract:</strong><br/>
We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ at sufficiently high temperatures whereas the spectral gap decays exponentially in $N$ at sufficiently low temperatures. These verify the existence of a dynamical high temperature phase and a dynamical glass phase at the level of the spectral gap. Key to these results are the understanding of the extremal process and restricted free energy of Subag–Zeitouni and Subag.
</p>projecteuclid.org/euclid.aihp/1557820830_20190514040050Tue, 14 May 2019 04:00 EDTExistence of Stein kernels under a spectral gap, and discrepancy boundshttps://projecteuclid.org/euclid.aihp/1557820831<strong>Thomas A. Courtade</strong>, <strong>Max Fathi</strong>, <strong>Ashwin Pananjady</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 777--790.</p><p><strong>Abstract:</strong><br/>
We establish existence of Stein kernels for probability measures on $\mathbb{R}^{d}$ satisfying a Poincaré inequality, and obtain bounds on the Stein discrepancy of such measures. Applications to quantitative central limit theorems are discussed, including a new central limit theorem in the Kantorovich–Wasserstein distance $W_{2}$ with optimal rate and dependence on the dimension. As a byproduct, we obtain a stable version of an estimate of the Poincaré constant of probability measures under a second moment constraint. The results extend more generally to the setting of converse weighted Poincaré inequalities. The proof is based on simple arguments of functional analysis.
Further, we establish two general properties enjoyed by the Stein discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is strictly decreasing along the CLT, and it controls the third moments of a random vector.
</p>projecteuclid.org/euclid.aihp/1557820831_20190514040050Tue, 14 May 2019 04:00 EDTHausdorff dimension of the scaling limit of loop-erased random walk in three dimensionshttps://projecteuclid.org/euclid.aihp/1557820832<strong>Daisuke Shiraishi</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 791--834.</p><p><strong>Abstract:</strong><br/>
Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in ( Ann. Probab. 46 (2) (2018) 687–774) that there exists $\beta\in(1,\frac{5}{3}]$ such that $E(M_{n})$ is of order $n^{\beta}$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $\beta$ almost surely.
</p>projecteuclid.org/euclid.aihp/1557820832_20190514040050Tue, 14 May 2019 04:00 EDTCondensation of a self-attracting random walkhttps://projecteuclid.org/euclid.aihp/1557820833<strong>Nathanaël Berestycki</strong>, <strong>Ariel Yadin</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 835--861.</p><p><strong>Abstract:</strong><br/>
We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $\beta $. We prove that, for all $\beta >0$, the random walk condensates to a set of diameter $(t/\beta )^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $d\ge 3$, we also prove that the volume is bounded above by $(t/\beta )^{d/(d+1)}$ and the diameter is bounded below by $(t/\beta )^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $\beta $ everywhere in its range when $\beta $ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.
</p>projecteuclid.org/euclid.aihp/1557820833_20190514040050Tue, 14 May 2019 04:00 EDTThe speed of biased random walk among random conductanceshttps://projecteuclid.org/euclid.aihp/1557820834<strong>Noam Berger</strong>, <strong>Nina Gantert</strong>, <strong>Jan Nagel</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 862--881.</p><p><strong>Abstract:</strong><br/>
We consider biased random walk among iid, uniformly elliptic conductances on $\mathbb{Z}^{d}$, and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1.1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 1.3: it follows along the lines of the proof of the Einstein relation in ( Ann. Probab. 45 (4) (2017) 2533–2567). On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if $d=2$ and if the conductances take the values $1$ (with probability $p$) and $\kappa$ (with probability $1-p$) and $p$ is close enough to $1$ and $\kappa$ small enough, the velocity is not increasing as a function of the bias, see Theorem 1.2.
</p>projecteuclid.org/euclid.aihp/1557820834_20190514040050Tue, 14 May 2019 04:00 EDTHow round are the complementary components of planar Brownian motion?https://projecteuclid.org/euclid.aihp/1557820835<strong>Nina Holden</strong>, <strong>Şerban Nacu</strong>, <strong>Yuval Peres</strong>, <strong>Thomas S. Salisbury</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 882--908.</p><p><strong>Abstract:</strong><br/>
Consider a Brownian motion $W$ in $\mathbf{C}$ started from $0$ and run for time 1. Let $A(1),A(2),\ldots$ denote the bounded connected components of $\mathbf{C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\mathbf{N}$. Our main result is that ${\mathbf{E}}[\sum_{i}R(i)^{2}|\log R(i)|^{\theta}]<\infty$ for any $\theta<1$. We also prove that $\sum_{i}r(i)^{2}|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.
</p>projecteuclid.org/euclid.aihp/1557820835_20190514040050Tue, 14 May 2019 04:00 EDTMatrix Dirichlet processeshttps://projecteuclid.org/euclid.aihp/1557820836<strong>Songzi Li</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 909--940.</p><p><strong>Abstract:</strong><br/>
Matrix Dirichlet processes, in reference to their reversible measure, appear in a natural way in many different models in probability. Applying the language of diffusion operators and the theory of boundary equations, we describe Dirichlet processes on the matrix simplex and provide two models of matrix Dirichlet processes, which can be realized by various projections, through the Brownian motion on the special unitary group and also through Wishart processes.
</p>projecteuclid.org/euclid.aihp/1557820836_20190514040050Tue, 14 May 2019 04:00 EDTParabolic Anderson model with rough or critical Gaussian noisehttps://projecteuclid.org/euclid.aihp/1557820837<strong>Xia Chen</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 941--976.</p><p><strong>Abstract:</strong><br/>
This paper considers the parabolic Anderson equation
\[{\frac{\partial u}{\partial t}}={\frac{1}{2}}\Delta u+u{\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\partial x_{1}\cdots\partial x_{d}}}\] generated by a $(d+1)$-dimensional fractional noise with the Hurst parameter $\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})$. The existence/uniqueness, Feynman–Kac’s moment formula and the precise intermittency exponents are formulated in the case when some of $H_{1},\ldots,H_{d}$ are less than one half, and in the case when the Dalang’s condition
\[d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{is replaced by }d-\sum_{k=1}^{n}H_{j}=1.\] Some partial result is also achieved for the case when $H_{0}<1/2$ which brings insight on what to expect as the Gaussian noise is rough in time.
</p>projecteuclid.org/euclid.aihp/1557820837_20190514040050Tue, 14 May 2019 04:00 EDTA new approach to the existence of invariant measures for Markovian semigroupshttps://projecteuclid.org/euclid.aihp/1557820838<strong>Lucian Beznea</strong>, <strong>Iulian Cîmpean</strong>, <strong>Michael Röckner</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 977--1000.</p><p><strong>Abstract:</strong><br/>
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we fix a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we obtain a unifying generalization of different versions for Harris’ ergodic theorem which provides an answer to an open question of Tweedie. Also, we show that for a nonlinear SPDE on a Gelfand triple, the strict coercivity condition is sufficient to guarantee the existence of a unique invariant probability measure for the associated semigroup, once it satisfies a Harnack type inequality with power. A corollary of the main result shows that any uniformly bounded semigroup on $L^{p}$ possesses an invariant measure and we give some applications to sectorial perturbations of Dirichlet forms.
</p>projecteuclid.org/euclid.aihp/1557820838_20190514040050Tue, 14 May 2019 04:00 EDTStatistical physics on a product of treeshttps://projecteuclid.org/euclid.aihp/1557820839<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1001--1010.</p><p><strong>Abstract:</strong><br/>
Let $G$ be the product of finitely many trees $T_{1}\times T_{2}\times\cdots\times T_{N}$, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new.
We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.
</p>projecteuclid.org/euclid.aihp/1557820839_20190514040050Tue, 14 May 2019 04:00 EDTFinite-time singularity of the stochastic harmonic map flowhttps://projecteuclid.org/euclid.aihp/1557820840<strong>Antoine Hocquet</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1011--1041.</p><p><strong>Abstract:</strong><br/>
We investigate the influence of an infinite dimensional Gaussian noise on the bubbling phenomenon for the stochastic harmonic map flow $u(t,\cdot):\mathbb{D}^{2}\to\mathbb{S}^{2}$, from the two-dimensional unit disc onto the sphere. The diffusion term is assumed to have range one pointwisely in the tangent space $T_{u(t,x)}\mathbb{S}^{2}$, so that the noise preserves the 1-corotational symmetry of solutions. Under the assumption that its space-correlation is of trace class (in some appropriate Hilbert space), we prove that the noise generates blow-up with positive probability. This scenario happens no matter how we choose the initial data, provided it fulfills the latter symmetry assumption.
</p>projecteuclid.org/euclid.aihp/1557820840_20190514040050Tue, 14 May 2019 04:00 EDTTransversal fluctuations for a first passage percolation modelhttps://projecteuclid.org/euclid.aihp/1557820841<strong>Yuri Bakhtin</strong>, <strong>Wei Wu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1042--1060.</p><p><strong>Abstract:</strong><br/>
In 1996, Licea, Newman, and Piza proved that for a rather convoluted definition of the transversal fluctuation exponent in first passage percolation, that exponent is bounded below by $3/5$. In this paper we introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$, and prove the same estimate for a natural clean notion of the exponent.
</p>projecteuclid.org/euclid.aihp/1557820841_20190514040050Tue, 14 May 2019 04:00 EDTContinuous-state branching processes, extremal processes and super-individualshttps://projecteuclid.org/euclid.aihp/1557820842<strong>Clément Foucart</strong>, <strong>Chunhua Ma</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1061--1086.</p><p><strong>Abstract:</strong><br/>
The long-term behavior of flows of continuous-state branching processes are characterized through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to $+\infty$ in the supercritical case, and slow down the rate of extinction in the subcritical one.
</p>projecteuclid.org/euclid.aihp/1557820842_20190514040050Tue, 14 May 2019 04:00 EDTBayesian nonparametric analysis of Kingman’s coalescenthttps://projecteuclid.org/euclid.aihp/1557820843<strong>Stefano Favaro</strong>, <strong>Shui Feng</strong>, <strong>Paul A. Jenkins</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1087--1115.</p><p><strong>Abstract:</strong><br/>
Kingman’s coalescent is one of the most popular models in population genetics. It describes the genealogy of a population whose genetic composition evolves in time according to the Wright–Fisher model, or suitable approximations of it belonging to the broad class of Fleming–Viot processes. Ancestral inference under Kingman’s coalescent has had much attention in the literature, both in practical data analysis, and from a theoretical and methodological point of view. Given a sample of individuals taken from the population at time $t>0$, most contributions have aimed at making frequentist or Bayesian parametric inference on quantities related to the genealogy of the sample. In this paper we propose a Bayesian nonparametric predictive approach to ancestral inference. That is, under the prior assumption that the composition of the population evolves in time according to a neutral Fleming–Viot process, and given the information contained in an initial sample of $m$ individuals taken from the population at time $t>0$, we estimate quantities related to the genealogy of an additional unobservable sample of size $m^{\prime}\geq1$. As a by-product of our analysis we introduce a class of Bayesian nonparametric estimators (predictors) which can be thought of as Good–Turing type estimators for ancestral inference. The proposed approach is illustrated through an application to genetic data.
</p>projecteuclid.org/euclid.aihp/1557820843_20190514040050Tue, 14 May 2019 04:00 EDTComparing mixing times on sparse random graphshttps://projecteuclid.org/euclid.aihp/1557820844<strong>Anna Ben-Hamou</strong>, <strong>Eyal Lubetzky</strong>, <strong>Yuval Peres</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1116--1130.</p><p><strong>Abstract:</strong><br/>
It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time ${\mathbf{h}}^{-1}\log n$, where ${\mathbf{h}}$ is the asymptotic entropy for simple random walk on a Galton–Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton–Watson tree.
</p>projecteuclid.org/euclid.aihp/1557820844_20190514040050Tue, 14 May 2019 04:00 EDTAn isomorphism between branched and geometric rough pathshttps://projecteuclid.org/euclid.aihp/1557820845<strong>Horatio Boedihardjo</strong>, <strong>Ilya Chevyrev</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1131--1148.</p><p><strong>Abstract:</strong><br/>
We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay–Victoir [ J. Differential Equations 225 (2006) 103–133] as well as a canonical version of the Itô–Stratonovich correction formula of Hairer–Kelly [ Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 207–251]. Our construction is elementary and uses the property that the Grossman–Larson algebra is isomorphic to a tensor algebra.
We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.
</p>projecteuclid.org/euclid.aihp/1557820845_20190514040050Tue, 14 May 2019 04:00 EDTOn random walk on growing graphshttps://projecteuclid.org/euclid.aihp/1557820846<strong>Ruojun Huang</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1149--1162.</p><p><strong>Abstract:</strong><br/>
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple random walk on slowly growing graphs, upon knowing the volume and Cheeger constant of each graph. For much more specialized cases, we establish matching lower bounds, and deduce sufficient (weak) recurrence criteria. We also address recurrence directly in relation to a universality conjecture of ( Electron. J. Probab. 19 (2014) Article ID 106). We answer a related question of ( Ann. Probab. 39 (2011) 1161–1203, Problem 1.8) about “inhomogeneous merging” in the negative.
</p>projecteuclid.org/euclid.aihp/1557820846_20190514040050Tue, 14 May 2019 04:00 EDTOn the equivalence between some jumping SDEs with rough coefficients and some non-local PDEshttps://projecteuclid.org/euclid.aihp/1557820847<strong>Nicolas Fournier</strong>, <strong>Liping Xu</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1163--1178.</p><p><strong>Abstract:</strong><br/>
We study some jumping SDE and the corresponding Fokker–Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_{t})_{t\in[0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_{t})_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli ( J. Funct. Anal. 254 (2008) 109–153) concerning continuous SDEs and local PDEs.
</p>projecteuclid.org/euclid.aihp/1557820847_20190514040050Tue, 14 May 2019 04:00 EDTLocal fluctuations of critical Mandelbrot cascadeshttps://projecteuclid.org/euclid.aihp/1557820848<strong>Dariusz Buraczewski</strong>, <strong>Piotr Dyszewski</strong>, <strong>Konrad Kolesko</strong>. <p><strong>Source: </strong>Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55, Number 2, 1179--1202.</p><p><strong>Abstract:</strong><br/>
We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_{n}(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu(B_{n}(x))$ as $n\to\infty$.
</p>projecteuclid.org/euclid.aihp/1557820848_20190514040050Tue, 14 May 2019 04:00 EDT